Brief Encounter is based on a one-act play called Still Life (1936) that was written by British playwright and actor Sir Noël Coward.
Ireland had strict censorship laws in the 1940s and banned this movie at the time because it portrayed adultery in a sympathetic light. For more information, check out Irish Film Censorship: A Cultural Journey from Silent Cinema to Internet Pornography (2004), Kevin Rockett's meticulously researched book on the subject.
