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 period ending June 30, 2003 as filed with the
Securities and Exchange Commission on the date hereof (the "Report"), I, Peter
K. Stevenson, Chief Executive Officer of the Company, certify, pursuant to
U.S.C. 'SS' 1350, as adopted pursuant to ss. 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 as of June 30, 2003 (the last date of
               the period covered by the Report).


/s/ Peter K. Stevenson
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Peter K. Stevenson
Chief Executive Officer

Date:  August 20, 2003