After years of relative anonymity, Scotsman Brian Cox has become one of Hollywood's busiest actors. He takes a break from acting for Woody Allen to talk to Leo Benedictus

Barefoot, and with a slightly distracted air, Brian Cox opens his front door to London's hottest day of the year. His hair is dyed rust-brown - an adjustment for Woody Allen's top-secret project, which he's filming in London at the moment.

He wafts me inside. Cox's house, like its owner, is surprisingly small but full of character. A fresh script lies open on the back of a chair. There's a toddler's climbing frame in the garden, and a meditation room down the hall.

Cox is 58, but his current schedule would exhaust three teenagers. He's just finished a movie with the Farrelly brothers; last weekend, his film The Bourne Supremacy made a $53m splash at the US box office; his Agamemnon is still battling Brad Pitt's Achilles on screens around the world; he has a two-year-old son, who is currently with his wife in their new home in LA; and another baby is expected in October, by which time the Coxes hope to have moved house. He also works as a drama teacher and theatre actor. And last night he was up until 3 a.m. trying to get his computer to work.

"Do you want to get something to eat?" he asks me, friendly but a little weary. "Yes," I say. I was prepared for this. Cox had said in advance that we should go to the pub, so I resolved to line my stomach. "Matthew!" he shouts up to an invisible computer-mender somewhere in his souk-cum-rabbit-warren of a house. "I'll be about 45 minutes!"

Cox's celebrity might be the strangest in Hollywood. Not because he doesn't deserve it, but because of how he made his name - with a small role as Dr Hannibal Lecktor (sic) in Michael Mann's Manhunter (1986). The movie grossed only $8.6m in the US, but became a retrospective classic five years later when Anthony Hopkins's Dr Lecter unleashed a new bogeyman on the world. How many other screen actors made their name playing a character synonymous with someone else?

At the bar, I ask for a pint. Now it's Cox's turn. "Can I have a mineral water, please," he says, "with a little apple juice in the bottom?" He may now have made his mark on Hollywood - but Hollywood, clearly, has also left its mark on him.

I ask him why casting agents love him so much. "They didn't always, and they may not do again. You can't believe all this stuff. You can't go around saying, 'I am Sir Thing. I have now arrived at a status in my life where I am very special and should be treated specially.' That's all horseshit. It's all dust within seconds."

Cox knows how quickly things can go wrong. He was born in Dundee in 1946 into a poor Irish immigrant family. When his father, a weaver, died of cancer when he was nine, his mother suffered a breakdown from which she never fully recovered. He was subsequently raised by his aunt and sister while his mother languished in an institution, undergoing the horrors of 1950s psychiatric treatment.

Cox always wanted to be a movie star, so he left home at 15 to get cracking. He trained at the London Academy of Music and Dramatic Arts, and found plenty of work in theatre and television, but it was only in the 1990s that his occasional character parts began to coalesce into a film career.

"People always said everything would come later to me, and it was true," he says.

Has he seen an effect on his fanbase? "It's got bigger, especially in the past year because of X-Men 2 and Troy. They've only recently begun to put a name to my face, but now they know there's a person behind all those different people. When we went to the première of The Bourne Supremacy, I came out and there must have been more than 100 people, which was scary.

"I was signing and signing and signing, but I couldn't get out, so in the end I just had to say stop." He sounds quite weary of the adulation. "But it's fine, as long as you're gracious. They're funny people, but they're nice people. I don't have stalkers."

At a time that he describes as the most satisfying of his career, Cox has also started to overcome an old hang-up. "I've begun to view my own work again. I've finally accepted the fact that I'm getting old - or older - so I can look at myself." Why couldn't he look at himself before? "Remember when you see the Elephant Man, and you're going, oh but his nose . . . and his ear . . . and his arm . . ." His voice teeters somewhere between pity and horror. "That's how I feel about myself. I do feel that I'm the Elephant Man when I watch myself on screen. I crawled out of Rob Roy on all fours - and I was wearing a kilt at the time. Funnily enough, I don't mind my own voice, it's just seeing all these bits that stick out, like noses, or big lobes on my ears, or chins. I come across as a sort of creature, I never seem like a human being. A lot of it's vanity. I much prefer to bathe in my own view of myself, which is not necessarily the physical reality."

Cox's physical reality is notably short and fairly plump. He is no matinee idol, but no gargoyle either. The word "craggy" has followed him everywhere for the past 15 years. And yet, as we talk, he radiates confidence - the kind that takes time to accrue.

What quickly becomes clear is that all his views fit somewhere into a leftish continuum (he describes himself as a socialist), in which hard work and professionalism are the great heroic virtues. The quantity of his output, coupled with the range of roles he has taken on, testify to this. While specialising in villains (although he dislikes this simplification - "You have to develop a point of view"), Cox was equally memorable as Ed Norton's sympathetic father in 25th Hour and as the comic Dr Guggenheim in Rushmore.

Cox is in the zone now, emphasising every point with a flourish of his hand, acting out snatches of dialogue where necessary.

I ask how the Woody Allen movie is going. "It's going very fast," he says. "It should be finished quite soon. I'm only doing it really to work with Woody. I love his films."

And what is Allen like to work with? "I think he intimidates everybody because he trusts you to do your job; he doesn't give you vast notes. I think he knows what he wants, but he doesn't declare that. Woody will say," (at this point he switches on a very passable impression) "'You know, it would be nice if I could recognise some of my own words, but that's okay.' He's like Clint Eastwood - you're wrapping by one o'clock in the afternoon."

I ask what his part is like. "I play the father of the family, who's a kind of upper-class character, which is a bit of a stretch for me." And are you bad? "No, I'm not bad, actually. I'm magnanimous."

He ponders this for a moment, grumbling something about how we'll probably still find some way to see this role as another bad guy. "He does buy his son-in-law; I suppose that's the one thing he does do." Buy his son-in-law? "Well I'm not allowed to go into it, but it's an interesting story." Having a topic of conversation banned must be a particular strain for someone as garrulous as Cox.

"This film has the same themes as his others, but it's done through a different filter - an English family. It's a moral tale, that much I will say. It's more in the Crimes and Misdemeanors area of his work."

It is Allen's unshowy approach to showbusiness that Cox has found a love for in the US. "The thing I like about America is that it's very professional," he says. "I can't say I don't like certain aspects of English life, because I do, but I've always felt like an observer, or a visitor. I think Tony Hopkins has the same thing that I have, only in his case it's more chronic, and I think more painful. I'm not pained by it; I'm more amused by it."

Back at his house, Cox's computer is nearly fixed. He looks so settled here that it's hard to believe he plans to leave. "The truth of the matter is I'm not leaving," he says. "I'm just buying a house. I'll be as much here as ever. And I like doing the theatre here, much more than in America." Working, bringing up small children and travelling - it must be exhausting?

"It's only exhausting if you're exhausted. It's infinitely more exhausting for my wife than it is for me. And I'm very good at catnapping - I've been like that since I was a wee boy." ...