Remember those days of the 1980s and 1990s when going to see an “action thriller” meant an evening of fireballs, lame one-liners and the prospect of Arnie, Bruce or Sly swinging in under a chopper to bazooka the baddies to goddamn-lady hell; those steroid-fests where men were men, sidekicks were sidekicks and the only serious acting on show was from the furtive RSC thesp moonlighting as chief villain? Most were highly entertaining. But these days, the thought of the world genuinelybeing saved by a cartoon Nietzschean? It’s like Bruce’s grubby regulation wife-beater: somehow, it just doesn’t wash.

It’s the post-9/11 world that has done it, so much so that even James Bond has been retooled. Sadly, our intimacy with both the reality of terror and the responsibility of global interdependence demands a greater degree of credibility from current films. But what a boon for movie-goers: Blood Diamond, The Constant Gardener, Syriana, The Good Shepherd. We would seem to be in the midst of a new era of intelligent thrillers. In the next few weeks come the espionage flick Breach, with Ryan Phillippe, the Michael Mann-produced The Kingdom (Jamie Foxx as an FBI agent in Saudi Arabia), and Neil Jordan’s The Brave One, with Jodie Foster as a vigilante.

Clearly, top-notch acting talent no longer regards these roles as second-division outings. Crime films, too, such as The Departed or George Clooney’s impending Michael Clayton, delve into hitherto unplumbed depths of moral complexity. “You’d think audiences would be timid,” says Paul Greengrass, the British director of the imminent Bourne Ultimatum. “It’s like 24 on TV. People love it. It goes back to the early 1970s and the great thrillers of that time – Three Days of the Condor, The Parallax View.” Reese Wither-spoon, Jake Gyllenhaal and Meryl Streep will appear in the war-on-terror picture Rendition. Leonardo DiCaprio, clearly liking his new direction, has just announced Penetration with Ridley Scott. Two films about Alexander Litvinenko are in the works. “When you’re in troubled times and the stakes are high,” Greengrass adds, “cinema comes alive.”

Is there any character more emblematic of this new wave than Jason Bourne? Embodied by Matt Damon, loosely based on Robert Ludlum’s cold-war potboilers, our favourite amnesiac CIA operative has, over three films, embarked on a gripping game of cat and mouse with his would-be assassins. “He’s not about Prada, he’s not about women in bikinis; it’s not about gadgets, it’s not about CGI,” says Greengrass. For Bourne, the height of decadence is putting on a sweatshirt; his car chases are full of the sickening crunches familiar to anyone who has been in a scrape. Nor is it about a Blofeld – the enemy, as with many of Bourne’s contemporaries, now seems to be the system.

“A lot of the credit,” Damon says, “goes to Doug Liman [who directed the first film, before Greengrass took over]. In the first meeting I ever had with him, he said, ‘I want to make a character that’s about today,someone that is walking around in the world.’” Both The Bourne Identity (2002) and The Bourne Supremacy (2004) proved big critical and commercial hits, making a combined $500m at the box office. But it is the third film that is arguably the best – a breathless chase through Moscow, Paris, Madrid, Tangier, New York and Waterloo station, Bourne armed with a gizmo no more intricate than a mobile phone (albeit with a generous roaming tariff). Shot in Greengrass’s fast-cut, hand-held style – as used in his United 93, a skill honed on television’s World in Action – the film has opened hugely in the States. “The first $100m student movie,” as the director himself puts it. But you can smell the verisimilitude: 20 years ago, Greengrass ghost-wrote the controversial MI5 confessional book Spycatcher.

Perhaps a greater turn-up for the books is that it should be Damon who has emerged as the great action star of the 21st century. Throw in the aforementioned Syriana, The Departed and The Good Shepherd and he has become the automatic choice when it comes to smart, tense, au courant material. “Knowing The Bourne Ultimatum was offin the middle distance gave me this real chance to do the best scripts I read,” he explains. “Those movies didn’t look like they were going to be box-office hits, but they turned out much better than anyone thought.” He is being modest. Add the Ocean’s films, and it is Damon who has emerged as the most bankable screen actor of our time. Not Brad, not George, not Tom, not Johnny. “I don’t think he gets nearly enough credit,” enthuses Greengrass. “To my mind, he is the outstanding actor. He has an ability that only very few actors have: to create an iconic character. They’ll still fall in love with the Bourne trilogy in 20 years.”

Meeting Damon – in LA, a place he visits only to tub-thump for a movie – is an enjoyable experience. Convivial, witty and articulate (he was a Harvard undergraduate when his film career took off), he also comes with an apparently sterling work ethic, the kind that has caused Coppola, Scorsese, Spielberg, Soderbergh, De Niro and, to no less a degree, Greengrass to fall over each other to work with him. At 36, he still brims with the boyish charm that saw him burst onto the scene 10 years ago. Buff and tanned (he did a lot of his own Bourne stunts), he is just back from a surfing trip to Hawaii, a long way in body and soul from, say, the anonymous CIA automaton of The Good Shepherd or the toady energy analyst of Syriana. Part of his skill is in morphing into blank, often duplicitous, characters. It is aided by his low-key public profile. “There’s absolutely nothing to see here,” he jokes. Last in the tabloids at the height of Matt’n’Ben mania, he lives a quiet life in Miami with his wife, Luciana Barroso (a former bartender), and their 14-month-old daughter, Isabella.

It has taken a long time for Damon to find his way. After the early promise of roles in Courage Under Fire and The Rainmaker – and let’s not forget the most famous soldier in recent screen history, Private Ryan – Damon immediately went off-piste into Kevin Smith comedies and indie flops such as All the Pretty Horses. Not that he has entirely curbed his wayward instincts: viz Stuck on You and The Brothers Grimm. For a Hollywood trying to pitch him as a matinée idol, such things were not a welcome diversion. His Robert Redford film, The Legend of Bagger Vance, duly tanked.

There was no doubt that Damon could act. The underground poker drama Rounders was a tour de force, The Talented Mr Ripley won rave reviews. But the poor box office again merely confirmed to the studios that a would-be heart-throb should not be seen as a murderous homosexual. “Part of that’s by design,” explains Damon. “I wouldn’t want a persona that got latched onto, because I’d get forced into doing the same thing over and over again. Some of the best roles in my career have been when directors have taken advantage of what image there is of me – all-American or something – and then subverted that.” Ben Affleck, meanwhile, was doing Armageddon and Pearl Harbor: you knew which actor Tinseltown had put its shirt on.

In the film business, a career can disappear faster than you can say “Chris O’Donnell”. Five years ago, Damon wound up living in a flat over a sandwich shop in London’s Soho while appearing in the West End production of Kenneth Lonergan’s This Is Our Youth. The transatlantic calls from his agent did not spell promise. “The rose-coloured glasses came off,” he recalls. “If your movies make money, there’s a job waiting for you, and it’s really not any more complicated than that.”

Although it proved to be his salvation, Jason Bourne, on paper, did not seem it. The first film was a troubled production from which Brad Pitt had already walked. Damon was brought in as a ringer. “It wouldn’t surprise me to find out Brad was offered The Bourne Ultimatum before me,” he quips. “Usually, the scripts I get have got Brad’s fingerprints on them, or Leo’s. But, yeah, we were a year delayed in coming out: there had been serious fights between the director and the studio, four rounds of reshoots. The writer, Tony Gilroy, was the first in the history of Hollywood to arbitrate against himself so that he did notget sole credit.”

Even the second film, with Greengrass, was fraught with problems. “The ending was flawed, and everybody knew it,” Damon says. A key scene in which Bourne has an encounter with his good-cop handler, Pam Landy (Joan Allen), had originally been shot with Bourne incarcerated in a prison hospital, then escaping, miraculously, by bending open the bars on the window. “They chased me for two movies in order to put me in a room I could get out of? It was totally stupid, and test audiences knew it.” With the release date imminent, it was Damon who sat down and rewrote the scene as a telephone conversation, with Bourne looking down from a rooftop, urging, memorably: “Get some rest, Pam, you look tired.” “He’s an Academy Award-winning screenwriter,” reminds Greengrass, “an absolute shaper of this trilogy.”

I must confess that I still find Damon’s screenwriting career perplexing. In 1997, it had provided one of the great Cinderella tales: Damon and Affleck, the boyhood chums from Boston who had followed their Hollywood dream, hitting gold with their first script, Good Will Hunting. There was schadenfreude aplenty from jealous rivals who suggested the boys had had more than a little “help” in writing the thing. “That rumour was put out the week the ballots were cast for the Oscars,” Damon elaborates. “It was never meant to gain any traction. It was just meant to put doubt into people who were voting in their mind for that week. You know, it was really Bill Goldman; Ted Tally was another one they said wrote it.”

He jokes: “It relieves me of the burden of trying to win awards ever again.” But, that said, given a genuine talent (and overlooking his penning of the indie film Gerry, for Gus Van Sant), has he never sought to silence his critics by coming up with something else? “That’s nice of you to say so, and I do want to write again. But I think it’s more a matter of having something to say. We wrote the movie to get jobs as actors and that has happened for the past 10 years. We miss it, we want to write, but it’s a matter of finding the time.”

Damon and Affleck are still thick as thieves (“If I ever woke up with a dead hooker in my hotel room, Matt would be the first person I’d call,” as Affleck once put it). It is Affleck with whom Damon has been on holiday, ripping a few waves, only these days, with the families in tow – in Affleck’s case, his wife, Jennifer Garner, and their infant daughter. Damon reels off a few paternity tips. “He’s about six months ahead of me, but he’s been very helpful.” Writing aside, they want to act together again. “Our careers are on a really solid footing. I hope this next decade will provide a number of opportunities for us.”

That would suggest their slates are pretty full. Damon has just done small parts in Coppola’s Youth Without Youth and Lonergan’s Margaret. He intends to get a boxing flick off the ground, The Fighter, with Mark Wahlberg, and quashes the rumour that he is set to play a young Captain Kirk in a Star Trek film. There remains, too, the possibility of reprising his Good Shepherd role in De Niro’s mooted second CIA instalment, going from the building of the Berlin Wall in 1962 to its fall in 1989. “The last time I spoke to him, he was pretty optimistic about it,” says Damon. “He always envisaged it as a three-film series.”More immediately, there is anotherthriller, Soderbergh’s The Informant, about a price-fixing scandal at a Fortune 500 company.

He lifts my tape recorder from the coffee table and slides out from underneath it a hand-bound script, the words “Imperial Life in the Emerald City” inked along its spine. It’s the story of the chaos inside modern Baghdad, based on the book of that title by Rajiv Chandrasekaran. If the dates work out, he and Greengrass are itching to reunite on it.

It seems unlikely, however, that they will be doing another Bourne. “This guy’s search for his identity is over,” he stresses. “It was three movies and he was haunted by all these things, and that’s what drives the guy. Now he knows everything, so it would be a complete reinvention to start from there. There’s nothing left for him to remember.”

The Bourne Ultimatum opens on Friday

Explore Film

• Film Reviews

• DVD

• Oscars

• Bollywood

Editor's choice

• Desperados with avocados - a sort of Guy Ritchie film
• Live fast, die young: The 27 Club
• Madonna's 50 most memorable moments