UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                   FORM N-CSR

                   CERTIFIED SHAREHOLDER REPORT OF REGISTERED
                         MANAGEMENT INVESTMENT COMPANIES

Investment Company Act file number: 811-05987

Morgan Stanley New York Municipal Money Market Trust
               (Exact name of registrant as specified in charter)

1221 Avenue of the Americas, New York, New York 10020
     (Address of principal executive offices)                         (Zip code)

Ronald E. Robison
1221 Avenue of the Americas, New York, New York 10020
                     (Name and address of agent for service)

Registrant's telephone number, including area code: 212-762-4000

Date of fiscal year end: December 31, 2004

Date of reporting period: June 30, 2004

Item 1 - Report to Shareholders



Welcome, Shareholder:

In this report, you'll learn about how your investment in Morgan Stanley New
York Municipal Money Market Trust performed during the semiannual period. We
will provide an overview of the market conditions, and discuss some of the
factors that affected performance during the reporting period. In addition, this
report includes the Fund's financial statements and a list of Fund investments.

This material must be preceded or accompanied by a prospectus for the fund being
offered. Market forecasts provided in this report may not necessarily come to
pass. There is no assurance that the Fund will achieve its investment objective.
The Fund is subject to market risk, which is the possibility that market values
of securities owned by the Fund will decline and, therefore, the value of the
Fund's shares may be less than what you paid for them. Accordingly, you can lose
money investing in this Fund.


FUND REPORT

For the six-month period ended June 30, 2004

MARKET CONDITIONS

Perhaps the most significant event of the six-month review period happened on
its final day. On June 30, 2004, the Federal Open Market Committee (the "Fed")
raised the federal funds rate target by 25 basis points, to 1.25 percent. Prior
to that, the target rate had been at a multidecade low for a full year. Indeed,
rates across the money market yield curve had steadily trended lower for several
years. The Fed's change had been widely anticipated; at the end of the period
the market reflected investors' general expectation that rates would rise still
further in the coming months as a result of mounting inflationary pressures.

The municipal money markets for the period were largely mixed. At the outset of
the period, the U.S. economy remained weak and state governments were facing
severe budgetary difficulties. Across the board, municipalities sought ways to
reduce expenditures and boost revenue through a variety of tactics, including
new fees and taxes, refinancing of existing debt and, in some instances,
borrowing to fund deficits. The economic and investment environment both showed
marked improvement in the spring of 2004, when continued growth in employment
and corporate profits alike resulted in improving outlooks for municipal tax
revenues. The money market yield curve -- the difference between short- and
longer-term yields -- steepened significantly over the first six months of 2004,
reflecting the market's expectation of further strength in the U.S. economy.

PERFORMANCE ANALYSIS

As of June 30, 2004, Morgan Stanley New York Municipal Money Market Trust had
net assets of more than $62 million and an average portfolio maturity of 37
days. For the six-month period ended June 30, 2004, the Fund provided a total
return of 0.12 percent. For the seven-day period ended June 30, 2004, the Fund
provided an effective annualized yield and a current yield both of 0.53 percent,
while its 30-day moving average yield for June was 0.52 percent. Past
performance is no guarantee of future results.

We were largely cautious in managing the Fund's portfolio during the period,
preferring to avoid the one-year segment of the money market in anticipation of
further yield increases there. As a result, the Fund's portfolio ended the
period with a weighted average maturity somewhat lower than has historically
been the case at this point in the year. We chose to emphasize short-term
variable-rate paper and shorter maturities of tax-exempt commercial paper in
order to minimize the potential adverse effects of rising interest rates. Given
the budgetary challenges facing the state of New York, we remained highly
cautious in our security selection over the course of the review period.

2


<Table>
<Caption>
   PORTFOLIO COMPOSITION
                                                 
   Variable Rate Municipal Obligations                 83.6%
   Municipal Notes                                      8.4
   Tax-Exempt Commercial Paper                          8.0
</Table>

<Table>
<Caption>
   MATURITY SCHEDULE
                                                 
     1-30 Days                                         80.8%
    31-60 Days                                          -
    61-90 Days                                          6.4
    91-120 Days                                         4.8
   121+ Days                                            8.0
</Table>

Data as of June 30, 2004. Subject to change daily. All percentages for Portfolio
Composition and Maturity Schedule are as a percentage of total investments.
Provided for informational purposes only and should not be deemed a
recommendation to buy or sell the securities mentioned. Morgan Stanley is a
full-service securities firm engaged in securities trading and brokerage
activities, investment banking, research and analysis, financing and financial
advisory services.

INVESTMENT STRATEGY

THE FUND WILL INVEST IN HIGH QUALITY, SHORT-TERM SECURITIES THAT ARE NORMALLY
MUNICIPAL OBLIGATIONS THAT PAY INTEREST EXEMPT FROM FEDERAL AND NEW YORK INCOME
TAXES. THE FUND'S "INVESTMENT MANAGER," MORGAN STANLEY INVESTMENT ADVISORS INC.,
SEEKS TO MAINTAIN THE FUND'S SHARE PRICE AT $1.00. THE SHARE PRICE REMAINING
STABLE AT $1.00 MEANS THAT THE FUND WOULD PRESERVE THE PRINCIPAL VALUE OF YOUR
INVESTMENT.

AN INVESTMENT IN THE FUND IS NOT INSURED OR GUARANTEED BY THE FEDERAL DEPOSIT
INSURANCE CORPORATION OR ANY OTHER GOVERNMENT AGENCY. ALTHOUGH THE FUND SEEKS TO
PRESERVE THE VALUE OF YOUR INVESTMENT AT $1.00 PER SHARE, IT IS POSSIBLE TO LOSE
MONEY BY INVESTING IN THE FUND.

                                                                               3


Morgan Stanley New York Municipal Money Market Trust
PORTFOLIO OF INVESTMENTS - JUNE 30, 2004 (UNAUDITED)

<Table>
<Caption>
PRINCIPAL
AMOUNT IN                                                                COUPON    DEMAND
THOUSANDS                                                                RATE+     DATE*        VALUE
- ---------------------------------------------------------------------------------------------------------
                                                                                 
            New York Tax-Exempt Short-Term Variable Rate Municipal Obligations
            (83.9%)
            Jay Street Development Corporation,
 $1,000       Fiscal 2001 Ser A-1......................................   1.00%   07/08/04   $ 1,000,000
  1,600       Fiscal 2001 Ser A-3......................................   1.07    07/08/04     1,600,000
  1,900     Long Island Power Authority, Electric Ser 7 Subser 7B
              (MBIA)...................................................   1.00    07/08/04     1,900,000
  1,500     Long Island Power Authority, Electric System Subser 1B.....   1.08    07/01/04     1,500,000
  1,495     Metropolitan Transportation Authority, ROCs II-R Ser 263
              (FSA)....................................................   1.10    07/08/04     1,495,000
  2,000     Nassau County Industrial Development Agency, 1999 Cold
              Spring Harbor Laboratory.................................   1.08    07/01/04     2,000,000
  1,900     Nassau County Interim Finance Authority, Sales Tax Ser 2002
              B (FSA)..................................................   1.00    07/08/04     1,900,000
  1,600     New York City, Fiscal 1994 Ser H Subser H-3 (FSA)..........   1.03    07/01/04     1,600,000
  2,000     New York City Housing Development Corporation, James Tower
              Development 2002 Ser A...................................   1.00    07/08/04     2,000,000
            New York City Municipal Water Finance Authority,
  1,800       1994 Ser C (FGIC)........................................   1.06    07/01/04     1,800,000
    840       PUTTERs Ser 401..........................................   1.09    07/08/04       840,000
            New York City Transitional Finance Authority,
  1,300       Recovery Fiscal 2003 Ser 1 Subser 1D.....................   1.08    07/01/04     1,300,000
  1,200       Recovery Fiscal 2003 Ser 3 Subser 3E.....................   1.08    07/01/04     1,200,000
  1,500     New York State, Ser 2000 A.................................   1.05    10/07/04     1,500,000
            New York State Dormitory Authority,
  1,000       Cornell University Ser 1990 B............................   1.08    07/01/04     1,000,000
  2,000       Mental Health Services Facilities Ser 2003D-2D (Ambac)...   1.07    07/08/04     2,000,000
    750       The Metropolitan Museum of Art Ser A.....................   1.01    07/08/04       750,000
  1,500       The New York Public Library Ser 1998 B (MBIA)............   1.00    07/08/04     1,500,000
  3,000     New York State Energy Research & Development Authority,
              Long Island Lighting Co Ser 1997 A (AMT).................   1.06    07/08/04     3,000,000
  1,500     New York State Environmental Quality Ser 1998 G............   1.03    10/01/04     1,500,000
            New York State Housing Finance Agency,
  3,000       150 East 44th Street 2000 Ser A (AMT)....................   1.08    07/08/04     3,000,000
  3,200       750 Sixth Avenue 1998 Ser A (AMT)........................   1.06    07/08/04     3,200,000
  2,500       Historic Front Street 2003 Ser A.........................   1.05    07/08/04     2,500,000
  1,500     New York State Local Government Assistance Corporation, Sub
              Lien Ser 2003 A-7V (FGIC)................................   1.00    07/08/04     1,500,000
  1,500     New York State Power Authority, Ser 1985...................   0.95    09/01/04     1,500,000
  3,000     Oneida Indian Nation of New York, Ser 2002.................   1.03    07/08/04     3,000,000
  1,800     Suffolk County Water Authority, Ser 2003 BANs..............   1.07    07/08/04     1,800,000
  2,500     Triborough Bridge & Tunnel Authority, Ser 2002 F...........   1.06    07/08/04     2,500,000
  1,000     Yonkers Industrial Development Agency, Consumers Union Ser
              1989.....................................................   1.05    07/08/04     1,000,000
</Table>

4
                       See Notes to Financial Statements


Morgan Stanley New York Municipal Money Market Trust
PORTFOLIO OF INVESTMENTS - JUNE 30, 2004 (UNAUDITED) continued

<Table>
<Caption>
PRINCIPAL
AMOUNT IN                                                                COUPON    DEMAND
THOUSANDS                                                                RATE+     DATE*        VALUE
- ---------------------------------------------------------------------------------------------------------
                                                                                 
            Puerto Rico,
 $1,000     Commonwealth of Puerto Rico Public Improvement Bonds, Ser
              2001-1 TOCs (FSA)........................................   1.35%   12/02/04   $ 1,000,000
                                                                                             -----------
            Total New York Tax-Exempt Short-Term Variable Rate Municipal Obligations (Cost
            $52,385,000)..................................................................    52,385,000
                                                                                             -----------
</Table>

<Table>
<Caption>
                                                                                 YIELD TO
                                                                                 MATURITY
                                                            COUPON   MATURITY   ON DATE OF
                                                             RATE      DATE      PURCHASE
                                                            ------   --------   ----------
                                                                              
            New York Tax-Exempt Commercial Paper (8.0%)
  1,000     New York State, Environmental Quality Ser 1997
              A...........................................   0.95%   07/20/04      0.95%       1,000,000
  1,500     New York State Dormitory Authority, Columbia
              University 1997 Issue.......................   1.08    07/28/04      1.08        1,500,000
            Puerto Rico
            Puerto Rico Government Development Bank,
  1,000       Ser 1986....................................   1.12    09/09/04      1.12        1,000,000
  1,500       Ser 1986....................................   1.15    09/14/04      1.15        1,500,000
                                                                                             -----------
            Total New York Tax-Exempt Commercial Paper
            (Cost $5,000,000).............................................................     5,000,000
                                                                                             -----------
            New York Tax-Exempt Short-Term Municipal Notes (8.5%)
    250     Islip, Public Improvement Ser 1995 (FGIC), dtd
              07/15/95....................................   5.00    07/15/04      1.00          250,381
  1,500     Oyster Bay, 2004 Ser A TANs, dtd 01/23/04.....   1.90    01/21/05      1.10        1,506,607
  1,000     Spencer-Van Etten Central School District, Ser
              2004 BANs, dtd 06/17/04.....................   3.00    06/17/05      1.88        1,010,616
  1,500     Steuben-Allegany Board of Cooperative
              Services, Sole Supervisory District RANs Ser
              2004, dtd 06/29/04..........................   3.00    06/30/05      1.78        1,517,921
            Puerto Rico
  1,000     Puerto Rico, Ser 2004 TRANs, dtd 10/15/03.....   2.00    07/30/04      1.00        1,000,788
                                                                                             -----------
            Total New York Tax-Exempt Short-Term Municipal Notes
            (Cost $5,286,313).............................................................     5,286,313
                                                                                             -----------
</Table>

                                                                               5
                       See Notes to Financial Statements


Morgan Stanley New York Municipal Money Market Trust
PORTFOLIO OF INVESTMENTS - JUNE 30, 2004 (UNAUDITED) continued

<Table>
<Caption>
                                                                                                VALUE
- ---------------------------------------------------------------------------------------------------------
                                                                                    
            Total Investments
            (Cost $62,671,313) (a)............................................    100.4%     $62,671,313

            Liabilities in Excess of Other Assets.............................    (0.4)         (245,669)
                                                                                  -----      -----------
            Net Assets........................................................    100.0%     $62,425,644
                                                                                  =====      ===========
</Table>

- ---------------------

<Table>
            
     AMT       Alternative Minimum Tax.
    BANs       Bond Anticipation Notes.
   PUTTERs     Puttable Tax-Exempt Receipts.
    RANs       Revenue Anticipation Notes.
    ROCs       Reset Option Certificates.
    TANs       Tax Anticipation Notes.
    TOCs       Tender Option Certificates.
    TRANs      Tax and Revenue Anticipation Notes.
      +        Rate shown is the rate in effect at June 30, 2004.
      *        Date on which the principal amount can be recovered through
               demand.
     (a)       Cost is the same for federal income tax purposes.

Bond Insurance:
- ---------------
    Ambac      Ambac Assurance Corporation.
     FSA       Financial Security Assurance Inc.
    FGIC       Financial Guaranty Insurance Company.
    MBIA       Municipal Bond Investors Assurance Corporation.
</Table>

6
                       See Notes to Financial Statements


Morgan Stanley New York Municipal Money Market Trust
FINANCIAL STATEMENTS

Statement of Assets and Liabilities
June 30, 2004 (unaudited)

<Table>
                                                           
Assets:
Investments in securities, at value
  (cost $62,671,313)........................................  $62,671,313
Cash........................................................       74,434
Interest receivable.........................................      116,392
Prepaid expenses and other assets...........................       29,813
                                                              -----------
    Total Assets............................................   62,891,952
                                                              -----------
Liabilities:
Payable for:
    Shares of beneficial interest redeemed..................      360,000
    Investment management fee...............................       11,296
    Distribution fee........................................        5,730
Accrued expenses and other payables.........................       89,282
                                                              -----------
    Total Liabilities.......................................      466,308
                                                              -----------
    Net Assets..............................................  $62,425,644
                                                              ===========
Composition of Net Assets:
Paid-in-capital.............................................  $62,425,091
Accumulated undistributed net investment income.............          553
                                                              -----------
    Net Assets..............................................  $62,425,644
                                                              ===========
Net Asset Value Per Share,
62,425,618 shares outstanding (unlimited shares authorized
of $.01 par value)..........................................        $1.00
                                                              ===========
</Table>

                                                                               7
                       See Notes to Financial Statements


Morgan Stanley New York Municipal Money Market Trust
FINANCIAL STATEMENTS continued

Statement of Operations
For the six months ended June 30, 2004 (unaudited)

<Table>
                                                           
Net Investment Income:
Interest Income.............................................  $327,511
                                                              --------
Expenses
Investment management fee...................................   165,611
Professional fees...........................................    39,930
Distribution fee............................................    33,122
Transfer agent fees and expenses............................    17,243
Shareholder reports and notices.............................    15,499
Trustees' fees and expenses.................................     3,916
Custodian fees..............................................     3,363
Registration fees...........................................     2,990
Other.......................................................     2,475
                                                              --------
    Total Expenses..........................................   284,149
Less: amounts waived........................................   (31,824)
Less: expense offset........................................    (3,150)
                                                              --------
    Net Expenses............................................   249,175
                                                              --------
Net Investment Income.......................................  $ 78,336
                                                              ========
</Table>

8
                       See Notes to Financial Statements


Morgan Stanley New York Municipal Money Market Trust
FINANCIAL STATEMENTS continued

Statement of Changes in Net Assets

<Table>
<Caption>
                                                               FOR THE SIX      FOR THE YEAR
                                                              MONTHS ENDED          ENDED
                                                              JUNE 30, 2004   DECEMBER 31, 2003
                                                              -------------   -----------------
                                                               (unaudited)
                                                                        
Increase (Decrease) in Net Assets:
Operations:
Net investment income.......................................   $    78,336      $    197,460
Dividends to shareholders from net investment income........       (78,376)         (197,469)
Net decrease from transactions in shares of beneficial
  interest..................................................    (5,660,298)      (15,398,751)
                                                               -----------      ------------
    Net Decrease............................................    (5,660,338)      (15,398,760)
Net Assets:
Beginning of period.........................................    68,085,982        83,484,742
                                                               -----------      ------------
End of Period
(Including accumulated undistributed net investment income
of $553 and $593, respectively).............................   $62,425,644      $ 68,085,982
                                                               ===========      ============
</Table>

                                                                               9
                       See Notes to Financial Statements


Morgan Stanley New York Municipal Money Market Trust
NOTES TO FINANCIAL STATEMENTS - JUNE 30, 2004 (UNAUDITED)

1. Organization and Accounting Policies

Morgan Stanley New York Municipal Money Market Trust (the "Fund") is registered
under the Investment Company Act of 1940, as amended (the "Act"), as a
non-diversified, open-end management investment company. The Fund's investment
objective is to provide a high level of daily income which is exempt from
federal and New York income tax, consistent with stability of principal and
liquidity. The Fund was organized as a Massachusetts business trust on December
28, 1989 and commenced operations on March 20, 1990.

The following is a summary of significant accounting policies:

A. Valuation of Investments -- Portfolio securities are valued at amortized
cost, which approximates market value.

B. Accounting for Investments -- Security transactions are accounted for on the
trade date (date the order to buy or sell is executed). Realized gains and
losses on security transactions are determined by the identified cost method.
Discounts are accreted and premiums are amortized over the life of the
respective securities. Interest income is accrued daily.

C. Federal Income Tax Policy -- It is the Fund's policy to comply with the
requirements of the Internal Revenue Code applicable to regulated investment
companies and to distribute substantially all of its taxable and nontaxable
income to its shareholders. Accordingly, no federal income tax provision is
required.

D. Dividends and Distributions to Shareholders -- The Fund records dividends and
distributions to shareholders as of the close of each business day.

E. Use of Estimates -- The preparation of financial statements in accordance
with generally accepted accounting principles requires management to make
estimates and assumptions that affect the reported amounts and disclosures.
Actual results could differ from those estimates.

2. Investment Management Agreement

Pursuant to an Investment Management Agreement with Morgan Stanley Investment
Advisors Inc. (the "Investment Manager"), the Fund pays the Investment Manager a
management fee, accrued daily and payable monthly, by applying the following
annual rates to the net assets of the Fund determined as of the close of each
business day: 0.50% to the portion of the daily net assets not exceeding $500
million; 0.425% to the portion of the daily net assets exceeding $500 million
but not exceeding $750 million; 0.375% to the portion of the daily net assets
exceeding $750 million but not exceeding $1 billion; 0.35% to the portion of the
daily net assets exceeding $1 billion but not exceeding $1.5 billion; 0.325% to
the portion of the daily net assets exceeding $1.5 billion but not

10


Morgan Stanley New York Municipal Money Market Trust
NOTES TO FINANCIAL STATEMENTS - JUNE 30, 2004 (UNAUDITED) continued

exceeding $2 billion; 0.30% to the portion of the daily net assets exceeding $2
billion but not exceeding $2.5 billion; 0.275% to the portion of the daily net
assets exceeding $2.5 billion but not exceeding $3 billion; and 0.25% to the
portion of daily net assets exceeding $3 billion.

Effective May 1, 2004, the Investment Manager has agreed to assume all operating
expenses and to waive the compensation provided for in its Investment Management
Agreement to the extent that such expenses and compensation on an annualized
basis exceed 0.60% of the daily net assets of the Fund.

For the period January 1, 2004 through April 30, 2004, the Investment Manager
voluntarily waived $1,199 of its fee.

3. Plan of Distribution

Morgan Stanley Distributors Inc. (the "Distributor"), an affiliate of the
Investment Manager, is the distributor of the Fund's shares and in accordance
with a Plan of Distribution (the "Plan") pursuant to Rule 12b-1 under the Act,
finances certain expenses in connection with the promotion of sales of Fund
shares.

Reimbursements for these expenses are made in monthly payments by the Fund to
the Distributor, which will in no event exceed an amount equal to a payment at
the annual rate of 0.15% of the Fund's average daily net assets during the
month. Expenses incurred by the Distributor pursuant to the Plan in any fiscal
year will not be reimbursed by the Fund through payments accrued in any
subsequent fiscal year. For the six months ended June 30, 2004, the distribution
fee was accrued at the annual rate of 0.10%.

4. Security Transactions and Transactions with Affiliates

The cost of purchases and proceeds from sales/maturities of portfolio securities
for the six months ended June 30, 2004, aggregated $85,626,200 and $91,545,000,
respectively.

Morgan Stanley Trust, an affiliate of the Investment Manager and Distributor, is
the Fund's transfer agent. At June 30, 2004, the Fund had transfer agent fees
and expenses payable of approximately $660.

The Fund has an unfunded noncontributory defined benefit pension plan covering
certain independent Trustees of the Fund who will have served as independent
Trustees for at least five years at the time of retirement. Benefits under this
plan are based on factors which include years of service and compensation.
Aggregate pension costs for the six months ended June 30, 2004, included in
Trustees' fees and expenses in the Statement of Operations amounted to $3,479.
At June 30,

                                                                              11


Morgan Stanley New York Municipal Money Market Trust
NOTES TO FINANCIAL STATEMENTS - JUNE 30, 2004 (UNAUDITED) continued

2004, the Fund had an accrued pension liability of $57,287 which is included in
accrued expenses in the Statement of Assets and Liabilities. On December 2,
2003, the Trustees voted to close the plan to new participants and eliminate the
future benefits growth due to increases to compensation after July 31, 2003.

Effective April 1, 2004, the Fund began an unfunded Deferred Compensation Plan
(the "Compensation Plan") which allows each independent Trustee to defer payment
of all, or a portion, of the fees he receives for serving on the Board of
Trustees. Each eligible Trustee generally may elect to have the deferred amounts
credited with a return equal to the total return on one or more of the Morgan
Stanley funds that are offered as investment options under the Compensation
Plan. Appreciation/depreciation and distributions received from these
investments are recorded with an offsetting increase/decrease in the deferred
compensation obligation and do not affect the net asset value of the Fund.

5. Shares of Beneficial Interest

Transactions in shares of beneficial interest, at $1.00 per share, were as
follows:

<Table>
<Caption>
                                                               FOR THE SIX         FOR THE
                                                              MONTHS ENDED       YEAR ENDED
                                                              JUNE 30, 2004   DECEMBER 31, 2003
                                                              -------------   -----------------
                                                               (unaudited)
                                                                        
Shares sold.................................................    55,970,693       165,036,737
Shares issued in reinvestment of dividends..................        78,376           197,469
                                                               -----------      ------------
                                                                56,049,069       165,234,206
Shares redeemed.............................................   (61,709,367)     (180,632,957)
                                                               -----------      ------------
Net decrease in shares outstanding..........................    (5,660,298)      (15,398,751)
                                                               ===========      ============
</Table>

6. Expense Offset

The expense offset represents a reduction of the custodian fees for earnings on
cash balances maintained by the Fund.

7. Legal Matter

The Investment Manager, certain affiliates of the Investment Manager, certain
officers of such affiliates and certain investment companies advised by the
Investment Manager or its affiliates, including the Fund, are named as
defendants in a number of similar class action complaints which were recently
consolidated. This consolidated action also names as defendants certain
individual Trustees and Directors of the Morgan Stanley funds. The consolidated
amended complaint generally alleges that defendants, including the Fund,
violated their statutory disclosure obligations and

12


Morgan Stanley New York Municipal Money Market Trust
NOTES TO FINANCIAL STATEMENTS - JUNE 30, 2004 (UNAUDITED) continued

fiduciary duties by failing properly to disclose (i) that the Investment Manager
and certain affiliates of the Investment Manager allegedly offered economic
incentives to brokers and others to recommend the funds advised by the
Investment Manager or its affiliates to investors rather than funds managed by
other companies, and (ii) that the funds advised by the Investment Manager or
its affiliates, including the Fund, allegedly paid excessive commissions to
brokers in return for their efforts to recommend these funds to investors. The
complaint seeks, among other things, unspecified compensatory damages,
rescissionary damages, fees and costs. The defendants have moved to dismiss the
action and intend to otherwise vigorously defend it. While the Fund believes
that it has meritorious defenses, the ultimate outcome of this matter is not
presently determinable at this early stage of the litigation, and no provision
has been made in the Fund's financial statements for the effect, if any, of this
matter.

                                                                              13


Morgan Stanley New York Municipal Money Market Trust
FINANCIAL HIGHLIGHTS

Selected ratios and per share data for a share of beneficial interest
outstanding throughout each period:

<Table>
<Caption>
                                            FOR THE SIX                       FOR THE YEAR ENDED DECEMBER 31,
                                           MONTHS ENDED       ----------------------------------------------------------------
                                           JUNE 30, 2004        2003          2002          2001          2000          1999
                                           -------------      --------      --------      --------      --------      --------
                                            (unaudited)
                                                                                                    

Selected Per Share Data:

Net asset value, beginning of period.....      $ 1.00          $ 1.00        $ 1.00        $ 1.00        $ 1.00       $  1.00
                                               ------          ------        ------        ------        ------       -------

Net income from investment operations....       0.001           0.003         0.006         0.019         0.031         0.023

Less dividends from net investment
 income..................................      (0.001)         (0.003)       (0.006)       (0.019)       (0.031)       (0.023)
                                               ------          ------        ------        ------        ------       -------

Net asset value, end of period...........      $ 1.00          $ 1.00        $ 1.00        $ 1.00        $ 1.00       $  1.00
                                               ======          ======        ======        ======        ======       =======

Total Return.............................        0.12%(1)        0.27%         0.64%         1.90%         3.15%         2.29%

Ratios to Average Net Assets:

Expenses (before expense offset).........        0.76%(2)(3)(4)    0.79%(3)(4)    0.80%(3)    0.78%(3)     0.82%(3)      0.88%(3)

Net investment income....................        0.24%(2)(4)     0.27%(4)      0.63%         1.90%         3.12%         2.25%

Supplemental Data:
Net assets, end of period, in
 thousands...............................     $62,426         $68,086       $83,485       $75,240       $73,250       $62,009
</Table>

- ---------------------

<Table>
                                                      
    (1)  Not annualized.
    (2)  Annualized.
    (3)  Does not reflect the effect of the expense offset of 0.01%.
    (4)  If the Investment Manager had not waived a portion of its fee, the annualized expense and net investment income
         ratios would have been as follows:



                                   EXPENSE         NET INVESTMENT
           PERIOD ENDED:           RATIO           INCOME RATIO
         -----------------          ----                ----
         June 30, 2004              0.86%               0.14%
         December 31, 2003          0.80                0.26
</Table>

14
                       See Notes to Financial Statements


                      (This Page Intentionally Left Blank)


TRUSTEES

Michael Bozic
Charles A. Fiumefreddo
Edwin J. Garn
Wayne E. Hedien
James F. Higgins
Dr. Manuel H. Johnson
Joseph J. Kearns
Michael E. Nugent
Fergus Reid

OFFICERS

Charles A. Fiumefreddo
Chairman of the Board

Mitchell M. Merin
President

Ronald E. Robison
Executive Vice President and Principal Executive Officer

Barry Fink
Vice President

Joseph J. McAlinden
Vice President

Stefanie V. Chang
Vice President

Francis J. Smith
Treasurer and Chief Financial Officer

Thomas F. Caloia
Vice President

Mary E. Mullin
Secretary

TRANSFER AGENT

Morgan Stanley Trust
Harborside Financial Center, Plaza Two
Jersey City, New Jersey 07311

INDEPENDENT REGISTERED PUBLIC ACCOUNTING FIRM

Deloitte & Touche LLP
Two World Financial Center
New York, New York 10281

INVESTMENT MANAGER

Morgan Stanley Investment Advisors Inc.
1221 Avenue of the Americas
New York, New York 10020

The financial statements included herein have been taken from the records of the
Fund without examination by the independent auditors and accordingly they do not
express an opinion thereon.

This report is submitted for the general information of shareholders of the
Fund. For more detailed information about the Fund, its fees and expenses and
other pertinent information, please read its Prospectus. The Fund's Statement of
Additional Information contains additional information about the Fund, including
its trustees. It is available, without charge, by calling (800) 869-NEWS.

This report is not authorized for distribution to prospective investors in the
Fund unless preceded or accompanied by an effective Prospectus. Read the
Prospectus carefully before investing.

Investments and services offered through Morgan Stanley DW Inc., member SIPC.
Morgan Stanley Distributors Inc., member NASD.

(c) 2004 Morgan Stanley

[MORGAN STANLEY LOGO]

MORGAN STANLEY FUNDS

Morgan Stanley
New York Municipal
Money Market Trust

Semiannual Report
June 30, 2004

[MORGAN STANLEY LOGO]

                                                     37906RPT-RA04-00482P-Y06/04


Item 2.  Code of Ethics.

Not applicable for semiannual reports.

Item 3.  Audit Committee Financial Expert.

Not applicable for semiannual reports.

Item 4. Principal Accountant Fees and Services

Not applicable for semiannual reports.

Item 5. Audit Committee of Listed Registrants.

Not applicable for semiannual reports.

Item 6. [Reserved.]

Item 7. Disclosure of Proxy Voting Policies and Procedures for Closed-End
Management Investment Companies.

Not applicable for semiannual reports.

Item 8. [Reserved.]

Item 9 - Controls and Procedures

(a) The Fund's principal executive officer and principal financial officer have
concluded that the Fund's disclosure controls and procedures are sufficient to
ensure that information required to be disclosed by the Fund in this Form N-CSR
was recorded, processed, summarized and reported within the time periods
specified in the Securities and Exchange Commission's rules and forms, based
upon such officers' evaluation of these controls and procedures as of a date
within 90 days of the filing date of the report.

      There were no significant changes or corrective actions with regard to
significant deficiencies or material weaknesses in the Fund's internal controls
or in other factors that could significantly affect the Fund's internal controls
subsequent to the date of their evaluation.



(b) There were no changes in the registrant's internal control over financial
reporting that occurred during the registrant's most recent fiscal half-year
(the registrant's second fiscal half-year in the case of an annual report) that
has materially affected, or is reasonably likely to materially affect, the
registrant's internal control over financial reporting.

Item 10 Exhibits

(a) Code of Ethics - Not applicable for semiannual reports.

(b) A separate certification for each principal executive officer and principal
financial officer of the registrant are attached hereto as part of EX-99.CERT.

                                       2


                                   SIGNATURES

      Pursuant to the requirements of the Securities Exchange Act of 1934 and
the Investment Company Act of 1940, the registrant has duly caused this report
to be signed on its behalf by the undersigned, thereunto duly authorized.

Morgan Stanley New York Municipal Money Market Trust

/s/ Ronald E. Robison
Ronald E. Robison
Principal Executive Officer
August 19, 2004

      Pursuant to the requirements of the Securities Exchange Act of 1934 and
the Investment Company Act of 1940, this report has been signed by the following
persons on behalf of the registrant and in the capacities and on the dates
indicated.

/s/ Ronald E. Robison
Ronald E. Robison
Principal Executive Officer
August 19, 2004

/s/ Francis Smith
Francis Smith
Principal Financial Officer
August 19, 2004

                                       3