Martin made her Broadway debut in Leave It to Me!, featuring music and lyrics by Cole Porter and a book by Bella Spewack and Sam Spewack. Based on the Spewacks' play Clear All Wires, the musical told the story of an American ambassador to the Soviet Union who becomes homesick for Topeka, Kansas and tries to get recalled.
Leave It to Me! opened at the Imperial Theatre on Nov. 9, 1938, starring William Gaxton, Victor Moore and Sophie Tucker. Gene Kelly had a role as a dancer in the show, marking his first Broadway appearance.
The New York Times praised the musical's "swift, smart and ingenious" score, and Martin's performance of the second act show-stopper "My Heart Belongs to Daddy" turned her into an overnight sensation.
The musical premiered on Broadway on Apr. 7, 1949, at the Majestic Theatre. It was an immediate success, winning ten Tony Awards and the 1950 Pulitzer Prize for Drama, and went on to play 1,925 performances. At the time of its closing, it was the second longest-running musical in Broadway history, after Rodgers and Hammerstein's Oklahoma!
Martin received her first Tony Award for Best Actress in a Musical for her portrayal of Navy nurse Nellie Forbush. In his New York Times review, Brooks Atkinson called the musical "rhapsodically enjoyable" and described Martin's performance as "full of quicksilver, pertness and delight."
Her next major Broadway role came in Peter Pan, the musicalization of the classic James M. Barrie play, which opened on Oct. 20, 1954, at the Winter Garden Theatre. Martin starred as the title character opposite Cyril Ritchard as Captain Hook, Kathy Nolan as Wendy Darling and Margalo Gillmore as Mrs. Darling. The cast also featured Martin's 12-year-old daughter Heller Halliday as Liza.
Brooks Atkinson considered Martin to be "the liveliest Peter Pan in the record book," and her performance as the boy who wouldn't grow up earned her a Tony Award.
The production premiered at the Lunt-Fontanne Theatre on Nov. 16, 1959, and enjoyed a 1,443-performance run. Martin received her third Tony Award for Best Actress in a Musical, and the show tied with Fiorello! for Best Musical. In 1965, The Sound of Music was adapted into a movie musical starring Julie Andrews and Christopher Plummer.
In 1966, Martin starred opposite Robert Preston in the Tom Jones-Harvey Schmidt musical I Do! I Do!, based on Jan de Hartog's play The Fourposter. The two-character show spanned fifty years, focusing on the ups and downs of a married couple.
The production opened on Dec. 5, 1966, and ran for 560 performances at the 46th Street (now Richard Rodgers) Theatre. Martin was Tony-nominated for her portrayal of Agnes.