But now, as Correspondent Lesley Stahl reports, he's really stretching himself by playing the singer Bobby Darin in "Beyond the Sea," which opens next week.
In it, Spacey not only plays the lead role, he sings all the songs himself. He also wrote, directed and produced the movie. But it took Spacey more than a decade to get this movie made, since Hollywood didn't think it could be a big box-office success.
There was also the question of age. Was Spacey, 45, too old to play Bobby Darin, who died from heart problems at 37? "What I've tried to do in this is to capture his essence and his attack," says Spacey. "The way he approached a song, but it has to come from me."
Spacey, however, says he has more in common with Darin than just singing: "I guess the biggest thing that I recognize, empathize, am interested in exploring is the person – and I know I have this in myself – who wants to keep reinventing themselves."
Spacey is able to reinvent himself, but he also has an uncanny knack for mimicking other people, like legendary talk show host Johnny Carson, which he does for Stahl.
"There's just something about Johnny Carson that makes me smile," says Spacey, who's been living in London for more than a year now, as artistic director of the Old Vic Theatre.
Some of the world's greatest actors, past and present, have performed on the stage of this 187-year-old theater: Sir John Gielgud, Sir Laurence Olivier, Vivien Leigh, Katharine Hepburn, and Peter O'Toole.
"When O'Toole was here, I used to love to come and see him in a play called 'Jeffrey Bernard Is Unwell,'" recalls Spacey. "And one night, he knew I was there. As the second act starts, the curtain raises, and he's lit a fire in a bar. And he's trying to put the fire out with his coat. And he yelled up to me, 'Spacey, I hope you're learning something.'"
Spacey is unlike any artistic director that this venerable theater has ever seen. He drives his scooter around the streets of London, much as he did four years ago, when Stahl first interviewed him in New York City.
Back then, he was trying to lead an anonymous life in New York City, with his dogs, Mini and Legacy. "I've always believed the longer an actor remains in the shadows, then the characters emerge," says Spacey. "And that, for me, is my primary function."
Kevin Spacey Fowler was born in New Jersey, the youngest of three children. He attended high school in Southern California, performed in school plays, and honed his talent as an impersonator. He loved doing movie stars like Jack Lemmon and Jimmy Stewart.
After a brief stint in college, Spacey dropped Fowler from his name and moved to New York City, to study drama for two years at the Juilliard School – where he was remembered as talented, and a troublemaker.
"I just think I got a huge ego. I think in my case, I really wasn't very good at that point," says Spacey. "I just wasn't good enough to play in the leagues that I wanted to play in, and yet I was pretending like I was."
In 1986, Spacey heard that the play, "Long Day's Journey Into Night," was coming to Broadway with Jack Lemmon. When he couldn't get an audition, he went to a lecture given by the play's director, Jonathan Miller.
"There was an elderly woman who was asleep next to me, and sticking out of her purse was an invitation to a cocktail reception in honor of Dr. Jonathan Miller following this particular event," recalls Spacey.
"And I thought, 'You know, she's tired. She's not going to go because it's late. It was late. So I very carefully and very quietly lifted her invitation. … I stole a little old lady's invitation."
Once he got his foot in the door, he says he convinced Miller to let him try out for the role of Lemmon's son: "I did about four scenes with Jack, and at the end of it, Jack Lemmon walked up to me, and he put his hand on my shoulder and he says, 'You know what? I never thought I'd find a rotten kid, but you're it. Jesus Christ.'"
Spacey was already in his 30's when he started making it in the movies as a character actor. He played a string of dark, manipulative reprobates - including a ruthless movie producer and a slimy salesman - culminating with the creepy, smooth-talking Verbal Kint in "The Usual Suspects." That role earned him his first Academy Award for supporting actor.
"A part of acting forces you to spend time with yourself, examine qualities that a character might have that maybe you don't want to think you have, or maybe I haven't ever demonstrated them," says Spacey. "But they're within me."
He finally broke out of being typecast when he played Lester Burnham, the hapless husband having a mid-life crisis in "American Beauty." That role shattered any notion that Spacey couldn't be a leading man, and with it, he won his second Academy Award for best actor.
"Success is both a fantastic thing and a little dangerous," says Spacey. "And so, I think that one has to always try to keep success a little bit at bay."
Spacey has managed to still be a success by following up "American Beauty" with a series of films that some critics panned.
"I've heard the expression, 'The curse of the Oscar,'" says Stahl. "Is that what it is?"
"That only depends on how you measure success," says Spacey. "Right now, I'm running one of the greatest theaters in the English-speaking world. … As far as I'm concerned, [it's] some curse."
But even while Spacey was making movies, he kept acting on stage. In 1998, he was performing at the Old Vic when he learned that the theater might be getting a new owner.
"There was some talk about it being turned into a lap dancing club," says Spacey, laughing. "And don't think that we're not going to do a lap-dancing season. But that's, you know, that's something else."
Spacey's first production, which he directed, opened in September to lukewarm reviews, but audiences seem to love it. In May, Spacey will take another turn on stage as star in "The Philadelphia Story."
With his upcoming movie, "Beyond the Sea," Spacey says his mother, who died two years ago, kept pushing him to get the movie made. "I've been talking to my mom for years about it," he says.
"She thought this was the greatest idea ever. So the last year of her life, I sort of stopped everything, and every day, right up to the last few days, she would always lean over and she'd say, 'Give me a BD update' -- a Bobby Darin update. She wanted to know what was happening with the movie. And the film is for her."
"Beyond the Sea" premiered at the Toronto Film Festival last September to some good reviews. Spacey has made the movie he dreamed of making. But he says his job at the Old Vic is his real passion.
"Theater was not a stepping-stone to film," says Spacey. "Film has turned out more incredible, and given me a more opportunities than I ever imagined. But theater has always been it."
Does he have a formal commitment to stay as artistic director of the Old Vic?
"No. I mean, I guess they could kick me out tomorrow," says Spacey, laughing. "Or they can come, you know, when I'm taking my teeth out and going onstage as Polonius. What part am I playing? When do I go on?"