Exhibit 32.1


                            CERTIFICATION PURSUANT TO
                 18 U.S.C. SECTION 1350, AS ADOPTED PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002


         In connection with the Quarterly Report of Globix Corporation (the
"Company") on Form 10-Q for the fiscal quarter ending March 31, 2004 as filed
with the Securities and Exchange Commission (the "Report"), I, Peter K.
Stevenson, Chief Executive Officer of the Company, certify, pursuant to 18
U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley
Act of 2002, that:

         (1)  The Report fully complies with the requirements of Section 13(a)
              or 15(d) of the Securities Exchange Act of 1934; and

         (2)  The information contained in the Report fairly presents, in all
              material respects, the financial condition and results of
              operations of the Company.



Date: May 7, 2004
                                              /s/ Peter K. Stevenson
                                              ----------------------
                                              Peter K. Stevenson
                                              Chief Executive Officer



                                       25