Exhibit 23.1

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Consent of Independent Registered Public Accounting Firm

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The Board of Directors

Adaptimmune Therapeutics plc:

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We consent to the incorporation by reference in the registration statement (No. 333-212713) on Form S-3 of Adaptimmune Therapeutics plc of our reports dated February 25, 2021, with respect to the consolidated balance sheets of Adaptimmune Therapeutics plc as of December 31, 2020 and 2019, the related consolidated statements of operations, comprehensive loss, changes in equity, and cash flows for each of the years in the three-year period ended December 31, 2020, and the related notes (collectively, the “consolidated financial statements”), and the effectiveness of internal control over financial reporting as of December 31, 2020, which reports appear in the December 31, 2020 annual report on Form 10-K of Adaptimmune Therapeutics plc.

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Our report dated February 25, 2021 on the December 31, 2020 consolidated financial statements refers to a change in the method of accounting for leases as of January 1, 2019 due to the adoption of Accounting Standard Codification Topic 842, Leases.

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/s/ KPMG LLP

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Reading, United Kingdom

February 25, 2021