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 July 31, 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 July 31, 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: September 8, 2004