Cort was born in New Rochelle, New York, and grew up in Rye, New York. His father, Joseph Parker Cox, was a bandleader and pianist, as well as a World War II veteran and merchant. His mother, Alma Mary (née Court), was a reporter and a merchant, who also worked in MGM studios. Cort has four siblings—three younger sisters and one older brother. His parents ran a clothing business in downtown Rye from the 1950s until the mid-1980s.
Most of Cort's adolescence was spent caring for his sisters and father; his father had multiple sclerosis and died of it in 1971. He also engaged in reading and painting. As a teenager he was a local portrait painting prodigy and began taking acting lessons. He was educated in Catholic schools and graduated from Iona Preparatory School in New Rochelle in 1966.
Cox (renamed Cort, a variation of his mother's maiden name, as the name Wally Cox was already taken) was discovered in a revue by director Robert Altman, who subsequently cast him in two of his movies, MASH and Brewster McCloud, in which he played the title role. Cort next went on to his most famous role, as the suicide-obsessed Harold, in Harold and Maude. Though the film was not particularly successful at the time of its release, it later gained international cult status and now is acclaimed as an American film classic.
On Broadway, Cort appeared in the short-lived 1972 play Wise Child by Simon Gray. Cort was invited to live with the famous comedian Groucho Marx in his Bel-Air mansion, and was present at Marx's death in 1977.
In 1979, Cort nearly died in a car accident on the Hollywood Freeway where he collided with an abandoned car blocking a lane into which he was turning. He broke an arm and a leg and sustained a concussion and a fractured skull. His face was severely lacerated and his lower lip nearly severed. Years of plastic surgery, substantial hospital bills, a lost court case, and the disruption of his career ensued.
In 1989 he directed the "Hôtel de Paris" episode of the second series of the ZDF German television documentary series Hotels, about famous hotels around the world.
Cort has since appeared in a number of film, stage and TV roles: Endgame, He Who Gets Slapped, Sledge Hammer!, The Chocolate War, The Big Empty, Theodore Rex, Dogma, But I'm A Cheerleader, Pollock, The Twilight Zone, The Secret Diary of Sigmund Freud and The Life Aquatic With Steve Zissou. Cort lent his voice to Edgar the computer in the movie Electric Dreams.
Cort voiced Toyman, a Superman villain, over the course of various DCAU series including Superman: The Animated Series, Static Shock, and Justice League Unlimited, as well as the character Josiah Wormwood in an episode of the earlier DCAU production Batman: The Animated Series. He had a cameo appearance as himself in the Arrested Development episode "Fakin' It", hosting a daytime court show called Bud Cort, a competitor to a similar daytime court show in the series called Mock Trial with J. Reinhold.
On the November 8, 2007 episode of Ugly Betty, Cort made a guest appearance as the priest officiating at Wilhelmina Slater's ill-fated wedding. In 2010, he guest-starred on Criminal Minds in the episode "Mosley Lane." Cort played an elderly pedophile Roger Roycewood, who along with his wife, kidnapped and killed young children. In 2012, he appeared as the artist "Gleeko" in the episode "Exit Wound the Gift Shop" in the second season of Eagleheart.
Cort's voice can be heard as The King in the English-language version of The Little Prince (2015). The film premiered out of competition at the 2015 Cannes Film Festival and won the César Award for Best Animated Film in February 2016. It was made available to U.S. audiences through Netflix in 2016.