Martin was best known for co-writing the score for the MGM musical Meet Me In St. Louis, which earned him an Academy Award nomination for Best Song for "The Trolley Song". He received a Tony nomination for Best Score when the show was turned into a Broadway musical in 1989.
Martin also received three Tony nominations for his work on the 1964 musical High Spirits, adapted from Noel Coward's Blithe Spirit, on which he collaborated on the book, music, and lyrics.
Martin also wrote two other Broadway musicals, Make A Wish and Best Foot Forward. As a performer, he appeared in numerous Broadway musicals, including Hooray For What, and he provided vocal arrangements to a slew of other Broadway musicals, including The Boys from Syracuse, DuBarry Was a Lady, Gentlemen Prefer Blondes, Lorelei, Good News, and Sugar Babies. He also was the pianist for Judy Garland's legendary engagement at the Palace Theatre in 1951-1952.
Martin also received an Academy Award nomination for "Pass the Peace Pipe" from the 1947 film version of Good News.