Exhibit 32.1

CERTIFICATION OF CHIEF EXECUTIVE OFFICER*

PURSUANT TO SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

Pursuant to 18 U.S.C. Section 1350, as created by Section 906 of the Sarbanes-Oxley Act of 2002, the undersigned officer of Avery Dennison Corporation (the “Company”) hereby certifies, to the best of his knowledge, that:

(i) the Quarterly Report on Form 10-Q of the Company for the fiscal quarter ended March 27, 2004 (the “Report”) fully complies with the requirements of Section 13(a) or Section 15(d), as applicable, of the Securities Exchange Act of 1934, as amended; and

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

Dated: May 5, 2004

By:/s/ Philip M. Neal

Name: Philip M. Neal

Title: Chairman and Chief Executive Officer

* The above certification accompanies the issuer’s Quarterly Report on Form 10-Q and is furnished, not filed, as provided in SEC Release 33-8238 dated June 5, 2003.