Born in Brooklyn, the son of immigrant barber Abraham Greenwald, Kidd began studying modern dance while still in high school. After briefly studying chemical engineering at City College, he won a scholarship to the School of American Ballet and spent much of the 1940s as a soloist with Ballet Theater. He choreographed his first piece for the company, "On Stage!" in 1945, and left ballet for Broadway in 1947.
Kidd won a Tony for his first Broadway show as a choreographer, Finian's Rainbow 1947. His other Tonys came for Guys and Dolls 1951, Can-Can 1954, L'il Abner 1957 and Destry Rides Again 1960. His other Broadway credits as a choreographer and/or director included Wildcat 1960, Subways Are for Sleeping 1961, Breakfast at Tiffany's 1966, The Rothschilds 1972 and The Goodbye Girl 1993.
Kidd is survived by his second wife, Shelah Hackett, a former dancer he worked with on Broadway, and four children.