George Bell, the Bishop of Chichester, was instrumental in getting Eliot to work as writer with producer E. Martin Browne in producing the pageant play The Rock (1934). Bell then asked Eliot to write another play for the Canterbury Festival in 1935. Eliot agreed to do so if Browne once again produced (he did). The first performance was given on 15 June 1935 in the Chapter House of Canterbury Cathedral. Robert Speaight played the part of Becket. The production then moved to the Mercury Theatre, Notting Hill Gate in London and ran there for several months.
Other notable performances
Robert Donat played Becket at the Old Vic in 1953 in a production directed by Robert Helpmann.
Jonathan Frid played Becket off-Broadway in a production by Robert Teuscher for two weeks during March 1971 at Central Theatre, Park Ave., NYC, NY.
The RSC staged revivals in 1972 at the Aldwych Theatre, with Richard Pasco as Becket, and at The Swan in 1993 transferring to The Pit in 1994 with Michael Feast.
Little Spaniel Theatre staged a production in St Bartholemew-the-Great in Smithfield London in May and June 2014. 
Television and film
The play, starring Robert Speaight, was broadcast live on British television by the BBC in 1936 in its first few months of broadcasting TV.
The play was later made into a black and white film with the same title. It was directed by the Austrian director George Hoellering with music by the Hungarian composer Laszlo Lajtha and won the Grand Prix at the Venice Film Festival in 1951. It was released in the UK in 1952. In the film the fourth tempter is not seen. His voice was that of Eliot himself.
The play is the basis for the opera Assassinio nella cattedrale by the Italian composer Ildebrando Pizzetti, first performed at La Scala, Milan, in 1958.