It’s a roasting hot day when I meet John Turturro. Needless to say, we’re not in Britain, but sitting on the rooftop of a private members’ club in the middle of Cannes, where Turturro’s latest work, the touching familial drama Mia Madre, has just premiered at the film festival. Turturro, dressed today in a pale blue shirt and charcoal slacks, has experienced some of his greatest triumphs here – Best Actor in 1991 for Barton Fink, the Camera D’Or for his directorial debut Mac.

With tight-curly black hair, a prominent nose and a spiky nature, Turturro is not the sort to get too caught up in the hoopla. “At the end of the day, your job is to keep people awake. That’s your job. And then tell a story – if people like it, that’s what they remember. I’ve made movies that people love that never won anything, and they discovered later on, and that’s a joy. That’s what I always aspire towards. I always wanted to make a living doing interesting things.”

Now 58, Turturro could say ‘mission accomplished’; there aren’t many actors who can claim multiple movies for Martin Scorsese, Spike Lee and the Coen Brothers mixed alongside Hollywood blockbusters like Michael Bay’s Transformers and Adam Sandler comedies. He’s won an Emmy (for TV series Monk), an Obie (for off-Broadway play Danny And The Deep Blue Sea) and been nominated for a Golden Globe (for Robert Redford’s Quiz Show); only the Academy, in its infinite wisdom, has yet to recognize him at the Oscars.

‘Range’ is the keyword here. “Turturro has demonstrated enough quirky brilliance for us to hunt down his performances as once we did those of Peter Lorre or Warren Oates,” wrote critic David Thomson once. Capable of plumbing the depths (from his Italian racist in Do The Right Thing to his snivelling thug in Miller’s Crossing), he’s just as adept at the outrageously comic – think his bowler in The Big Lebowski, his butler in Sandler comedy Mr Deeds or even his Egyptian pharaoh in Ridley Scott’s recent Biblical epic Exodus: Gods And Kings.

For new movie Mia Madre, you might say he’s been in rehearsal his whole career. He plays Barry Huggins, a mildly insufferable film actor who has arrived in Rome to work on a film directed by Margherita Buy’s frazzled filmmaker, who must also contend with the ailing health of her mother. Turturro, speaking some Italian, is quite superb as the self-regarding Huggins, a life-and-soul-of-the-set character who can’t help but namedrop the time he worked with ‘Stanley’ (Kubrick, that is).

As much as Barry could be seen as the film’s clownish comic relief, he has moments of tenderness too; perfectly essayed by Turturro. So what does he make of Barry? He looks at me. “What do I make of him? I’ve seen him. I know him.” He sound wry. “These people are capable of all kinds of things; they take a job and they show up and they’re not really prepared to do it, and they’re under pressure, and then people freak out. I’ve seen people do all kinds of things: directors, actors, even cinematographers. It’s a crazy business.”

Like what? This sounds intriguing. “You see a lot of strange things,” nods Turturro, wary of being too indiscreet. “People being really afraid and fearful of working. The next day, they’re unemployed! I know great actors and sometimes they don’t work. You’re like, ‘Wow, they can’t get a job.’ So that does something to people; even people I know who are really, really successful, they’re like, ‘Wow, I’m not as successful as I was.’ So these are occupational hazards.”

Given that Turturro started in the early 1980s– his first on-screen appearance was as a non-speaking extra in Scorsese’s legendary Raging Bull – the film landscape has changed dramatically since then.

“I think earlier on in my life, I worked with [William] Friedkin [on To Live And Die In LA] and [Michael] Cimino [on The Sicilian] – that was a different world. Cocaine-driven, brutal, actors being handled different ways. It wasn’t very gentle, and I didn’t really like that. But I learned a lot; I learned to speak up for myself.”

It wasn’t until Spike Lee’s Do The Right Thing in 1989 that Turturro really commandeered attention on screen, but his debut decade had been kind, establishing long-lasting relationships with the likes of Woody Allen (after appearing in Hannah And Her Sisters), whom Turturro went on to direct in his own last film, 2013’s comedy Fading Gigolo. He also appeared for Scorsese as a pool shark in 1985’s The Color Of Money and met his “good friends” Joel and Ethan Coen some years before he made his debut for them in Miller’s Crossing.

Curiously, the Coens were the head of the Cannes jury this year. “I’m sure they got a kick out of Barry. They probably want to work with him!” They still talk about projects – like a sequel to Barton Fink. “In a business that you always meet new people, I don’t have any continuity, so it’s nice to have some relationships that last.” Likewise, he’s been with his wife, actress Katherine Borowitz, for the past 30 years; they have two children – Amedeo, 25, and Diego, 15. When it comes to other forms of collaboration, he and Borowitz have occasionally starred together – notably in the Macbeth aspired mob film Men Of Respect.

Born in Brooklyn, Turturro’s own family was artistic. His mother was a jazz singer with her brothers, while his own sibling Nicholas and cousin Aida are both actors (notably, in NYPD Blue and The Sopranos respectively). Still, his subsequent upbringing in Queens was rough; he learnt to box in his backyard with his father, a construction worker who had emigrated from Bari in Italy. “I’d hate to say there was a starkness to it,” he notes, “but there was certainly a vigorous approach to your daily encounters!”

In his spare time, he’d make scrapbooks of old-school movie stars like Burt Lancaster and Kirk Douglas; he’d also tell his mother he was sick so he could stay home from school – then he’d watch classics like White Heat, honing his impersonation of James Cagney. As he got older, he started performing these take-offs at weddings and banquets. He played sports too – baseball, in particular – but when he broke his hand, it sparked a re-think towards his love of performance.

Much of this was brought about by being raised around his father’s extended family, all these “outsized men”, as he puts it, in these blue-collar environments. “They were really big personalities, so it was always around me. It’s very odd. I have done a certain amount of classical theatre, and when I go back to doing that, I realize it’s more the milieu I’m from – because everything is bigger. It’s almost something I feel more comfortable with.”

After school, Turturro studied theatre at college in New York and then at Yale School of Drama. Early successes were also on stage – not least when he created the title role of the violent barfly in John Patrick Shanley’s Danny And The Deep Blue Sea (The New York Times called him “an astonishing newcomer”). But film was always hovering in his peripheral vision. When he starred with Jodie Foster in 1987’s crime yarn Five Corners, it was enough to convince Spike Lee to cast him in his breakout role, Do The Right Thing.

Playing a volatile Italian in that, Turturro’s links to the old country remain strong. “I think cultures nowadays with mass communication and mass marketing, everything gets flattened out. What makes people distinctive is their roots.” In 2009, he co-wrote and presented the documentary Rehearsal For A Sicilian Tragedy, exploring his maternal homeland of Sicily. He followed it by directing Passione, a documentary about the musical heritage of Naples. By 2011, he’d received his Italian passport and now has dual Italian and American citizenship.

It’s no wonder working in Italy with Nanni Moretti (winner of the 2001 Palme D'Or for The Son's Room) on Mia Madre appealed. “I’m sure I’ll do something else there,” he says. He’s next up in Hands Of Stone – a biopic about boxer Roberto Duran, starring Robert De Niro as his legendary trainer Ray Arcel. “I did a cameo for Bob De Niro, because he pestered me no end to do it!” They’ve worked together a bunch of times since Turturro’s fleeting appearance in Raging Bull. “I’m a Roberto Duran fan too… They sent me a message from Roberto Duran, and that solidified it. He said, ‘Please do my movie!’”

There’s also a reunion with Adam Sandler in the offing – a comedy western called The Ridiculous 6 – but Turturro wrinkles his nose when I suggest he’s adept at straddling indies and blockbusters. “I hate the idea that there’s this indie world. I hate to ghettoise,” he retorts. “When I was younger I just used to go to the movies and those big blockbuster movies were fewer and far between.”

Maybe it’s a sign of getting older. “If I was a young actor now, I really don’t know how interested I would be… to be an actor. They don’t make the kind of movies that really turn me on.”

Mia Madre opens in cinemas on September 25