Exhibit 32.1

Certification Pursuant to 18 U.S.C. Sec. 1350

In connection with the filing by Tremont Mortgage Trust (the “Company”) of the Annual Report on Form 10-K for the period ended December 31, 2018 (the “Report”), each of the undersigned hereby certifies, to the best of his knowledge:

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.

/s/ David M. Blackman	/s/ G. Douglas Lanois
David M. Blackman President and Chief Executive Officer	G. Douglas Lanois Chief Financial Officer and Treasurer

Date:    February 7, 2019