The English actor - who is best known for playing Dr Frank-N-Furter in Richard O'Brien and Jim Sharman's cult classic 1975 musical movie - died on Tuesday night at his Los Angeles home, TMZ reported.
Curry had been in ill health after having a serious stroke in 2012.
As well as his role in The Rocky Horror Picture Show, Curry's other most famous performances came as Pennywise the Dancing Clown in the 1990 TV miniseries It, based on Stephen King's novel of the same name, and as pompous Plaza Hotel concierge Mr Hector in Home Alone 2: Lost in New York, in which Macaulay Culkin reprised his role as Kevin McCallister in the 1992 sequel to Christmas comedy Home Alone.
Other notable roles came in Annie, The Three Musketeers, The Hunt for Red October, Clue - based on the popular murder mystery board game - Addams Family Reunion, Congo and Scary Movie 2.
He was nominated for the Laurence Olivier Award for Best Actor in a Musical for his roles on London's West End as The Pirate King in The Pirates of Penzance (1982) and King Arthur in Spamalot (2007).
His Broadway runs earned him three Tony Award nominations: his first for Best Actor in a Play for his role as Wolfgang Amadeus Mozart in Amadeus (1980), followed by two nominations for Best Featured Actor in a Play for My Favourite Year (1992) and Spamalot (2005).
In 2025, Curry released his memoir Vagabond.
At the end of last year, Curry opened up about his numerous health struggles after his stroke and also admitted he was "not scared" of dying.
Talking about his life after his stroke in 2012 - which left him unable to walk and reliant on a team of carers - during an appearance on CBS Sunday Morning, he said: "I don't fear death. I try to avoid it. I think we all do but I suspect that in the end, I will welcome it. I think it may be very comforting to go bye-bye, and I want to earn it."
He also confessed he was frightened when he first suffered a stroke because the same condition previously killed his father.
Curry said: "I was scared ... I had to learn how to speak again. That was very weird. I hated not being able to speak. (And) my face kind of went sideways."
The thespian previously revealed he suffered the stroke during a massage and his therapist was the one who noticed something was wrong and called an ambulance.
Speaking on stage at a special screening of The Rocky Horror Picture Show at the Academy Museum in Los Angeles last month, Curry told the audience: "I was having a massage at the time and I didn't even actually notice anything, but the guy who was doing the massage said, 'I'm worried about you, I want to call an ambulance.' And he did, and I said, 'That's so silly'."
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