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 on Form 10-Q of Gritstone Oncology, Inc. (the “Company”) for the period ended March 31, 2020, as filed with the Securities and Exchange Commission on the date hereof (the “Report”), Andrew Allen, M.D., Ph.D., President and Chief Executive Officer of the Company, and Jean-Marc Bellemin, Chief Financial Officer of the Company, respectively, hereby certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 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.

Date:	May 7, 2020	/s/ Andrew Allen
		Andrew Allen, M.D., Ph.D.
		President and Chief Executive Officer (Principal Executive Officer)
Date:	May 7, 2020	/s/ Jean-Marc Bellemin
		Jean-Marc Bellemin
		Chief Financial Officer (Principal Financial Officer)