[KAYE, SCHOLER, FIERMAN, HAYS & HANDLER, LLP LETTERHEAD]

                                                                     EXHIBIT 5.1

                                                                  (212) 836-8000

                                  May 27, 1999

Veeco Instruments Inc.
Terminal Drive
Plainview, New York 11803

Ladies and Gentlemen:

            We have acted as special counsel to Veeco Instruments Inc., a
Delaware corporation (the "Company"), in connection with the Company's
registration statement on Form S-8 (the "Registration Statement") to be filed
pursuant to the Securities Act of 1933, as amended. The Registration Statement
relates to an aggregate of 1,500,000 shares of the Company's common stock, par
value $.01 per share (the "Common Stock"), 1,400,000 of which may be issued upon
the exercise of stock options to be granted pursuant to the Company's Amended
and Restated 1992 Employees' Stock Option Plan, as amended to date, and 100,000
of which may be issued upon the exercise of stock options granted pursuant to
the Company's Amended and Restated 1994 Stock Option Plan for Outside Directors,
as amended to date (collectively, the "Option Plans").

            In that connection, we have reviewed the Company's certificate of
incorporation as amended, its by-laws, resolutions adopted by its Board of
Directors and its stockholders, the Registration Statement, the Option Plans and
such other documents and proceedings as we have deemed appropriate.

            On the basis of such review, and having regard to legal
considerations that we deem relevant, we are of the opinion that the shares of
Common Stock to be offered pursuant to the Registration Statement have been duly
authorized and, when issued in accordance with the terms set forth in the Option
Plans, will be validly issued, fully paid and nonassessable.

            We advise you that we are members only of the Bar of the State of
New York and that our opinion set forth above is based as to matters of law
solely on applicable provisions of the General Corporation Law of the State of
Delaware, and we express no opinions as to any other laws, statutes, ordinances,
rules or regulations.

            We hereby consent to the filing of this opinion as an exhibit to the
Registration Statement. In giving this opinion, we do not thereby admit that we
are within the category of


persons whose consent is required under Section 7 of the Act or the rules and
regulations of the Securities and Exchange Commission.

                                Very truly yours,


                                /s/ Kaye, Scholer, Fierman, Hays & Handler, LLP