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 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 November 3, 2018 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 November 3, 2018 fairly presents, in all material respects, the financial condition and results of operations of the Company.

/s/ Ernie Herrman

Name:  Ernie Herrman

Title:    Chief Executive Officer and President

Dated: December 4, 2018