He rose to prominence as Dr. Frank-N-Furter in the musical film adaptation of The Rocky Horror Show, reprising the role he had originated both in the West End and on Broadway.

His other notable film appearances include The Shout (1978), Times Square (1980), The Worst Witch (1986), The Hunt for Red October (1990), The Three Musketeers (1993), Congo (1995), Charlie's Angels (2000), Scary Movie 2 (2001), and Kinsey (2004).

Curry is also a prolific voice actor, with roles in animation including his Emmy Award-winning performance as Captain Hook on Peter Pan & the Pirates (1990–1991), Hexxus in the film FernGully: The Last Rainforest (1992), King Chicken in Duckman (1994–1997), Sir Nigel Thornberry in The Wild Thornberrys (1998–2004), and Chancellor Palpatine / Darth Sidious in Star Wars: The Clone Wars (2012–2014).

However, at the suggestion of director Sharman, the character evolved into the diabolical mad scientist and transvestite with an upper-class Belgravia accent.

"[12] This change carried over to the 1975 film adaptation, The Rocky Horror Picture Show,[13] which made Curry a household name and gave him a cult following.

In an interview with NPR, Curry called Rocky Horror a "rite of passage", and added that the film is "a guaranteed weekend party to which you can go with or without a date and probably find one if you don't have one, and it's also a chance for people to try on a few roles for size, you know?

[14] Shortly after the end of Rocky Horror's run on Broadway, Curry returned to the stage with Tom Stoppard's Travesties, which ran in London and New York from 1975 to 1976.

[19] In 1989, Curry returned once again to the New York stage in The Art of Success,[20] and in 1992 played Alan Swann in the Broadway version of My Favorite Year, earning him his second Tony Award nomination, this time for Best Performance by a Leading Actor in a Musical.

[21] In 2001, Curry appeared as Scrooge in a musical version of the Charles Dickens novella A Christmas Carol that played at Madison Square Garden.

[24] From May to August 2011, Curry was scheduled to portray the Player in a Trevor Nunn stage production of Tom Stoppard's Rosencrantz and Guildenstern Are Dead at the Chichester Festival Theatre and then in London.

Director Ridley Scott cast Curry in the film after watching him in Rocky Horror, thinking he was ideal to play the role of Darkness.

Ray Porter in Pass the Ammo, Dr. Thornton Poole in Oscar (1991), Mr. Hector the suspicious Plaza Hotel concierge in Home Alone 2: Lost in New York (1992), Jigsaw in Loaded Weapon 1 (1993) and as Long John Silver in Muppet Treasure Island.

Curry went on to play Thurman Rice, a supporting role in the biographical film Kinsey (2004) and portrayed Alexander Monro in the British black comedy Burke & Hare (2010).

[34] Curry had further roles in British television throughout the 1970s and 1980s, including the lead in Stephen Frears' 1975 adaptation of Three Men in a Boat, playing Bill Sikes in Oliver Twist (1982), playing aspiring actor-singer Larry Gormley in Blue Money (1985), and starring as the Grand Wizard in the children's Halloween film The Worst Witch (1986).

Curry has appeared in many animated television series and films, starting with the performance of the Serpent The Greatest Adventure: Stories from the Bible (1988).

Curry won a Daytime Emmy Award for his performance as Captain Hook in the Fox animated series Peter Pan and the Pirates (1990–1991).

Real Monsters (1994–1997), King Chicken in Duckman (1994–1997), Dr. Anton Sevarius in Gargoyles (1995–1996), Slagar in Redwall (2000–2001), Professor Finbarr Calamitous in The Adventures of Jimmy Neutron: Boy Genius (2003–2005), G. Gordon Godfrey in Young Justice (2012–2013), and The Sorcerer in Randy Cunningham: 9th Grade Ninja (2012–2014).

Curry has also done voice acting in a number of animated films, most notably FernGully: The Last Rainforest (1992), Don Bluth's The Pebble and the Penguin (1995), Beauty and the Beast: The Enchanted Christmas (1997) for which he received an Annie Award nomination, Scooby-Doo!

Curry's performance as Premier Anatoly Cherdenko in live-action cut scenes in Command & Conquer: Red Alert 3 (2008), have gone viral as a meme.

[36] Curry's audiobook work includes his Grammy-nominated narraton of Lemony Snicket's A Series of Unfortunate Events, Geraldine McCaughrean's Peter Pan in Scarlet, Charles Dickens' A Christmas Carol and the Abhorsen trilogy.

He has mentioned that his musical influences included jazz vocalists such as Billie Holiday and Louis Armstrong and he idolised the Beatles and the Rolling Stones as a teenager.

Highlights of the album are a reggae version of the Beatles' song "I Will", a rendition of "Wake Nicodemus" featuring the Pipes and Drums of the 48th Highlanders of Canada, and a bar-room ballad, "Alan", composed by Canadian singer-songwriter Tony Kosinec.

However, the album remained unreleased in its entirety until February 2010, when it was made available as a legal download entitled ...From the Vaults (though four tracks from these sessions had been released on a 1990 Rocky Horror box set).

In October 2020, Curry reprised his role as Dr. Frank-N-Furter in a live table reading of The Rocky Horror Picture Show in support of the Democratic Party of Wisconsin to raise funds for Joe Biden's presidential campaign.