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Doctor Who API
REST API for next-episode.net.
Catch The Show API is a free Video API that requires no API key, is served over HTTPS and has unconfirmed CORS support.

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Doctor Who API
Yes. The Catch The Show API is listed in the Video category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
No. The Catch The Show API requires no authentication, so you can start making requests without signing up.
CORS support for the Catch The Show API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The Catch The Show API is served over HTTPS, so it's safe to use in production.