EXHIBIT 32.1

                            CERTIFICATION PURSUANT TO
            SECTION 1350, CHAPTER 63 OF TITLE 18, UNITED STATES CODE,
                             AS ADOPTED PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

Pursuant to Section 1350, Chapter 63 of Title 18, United States Code, as adopted
pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, the undersigned, as
President and Chief Executive Officer of The TJX Companies, Inc. (the
"Company"), does hereby certify that to my knowledge:

      1.    the Company's Form 10-Q for the fiscal quarter ended October 30,
            2004 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 Company's Form 10-Q for the fiscal
            quarter ended October 30, 2004 fairly presents, in all material
            respects, the financial condition and results of operations of the
            Company.

                                        /s/ Edmond J. English
                                        ----------------------------------------
                                        Name:  Edmond J. English
                                        Title: President and Chief Executive
                                               Officer

Dated: December 9, 2004