Go to warp factor 10, Mr Chekhov

The melancholy Muscovite has gone into orbit, thanks to one theatre.

Theatres everywhere are desperate for them. The National has split its Lyttelton stage in two and slashed beer prices on their behalf. Will it work? Who knows how to woo young audiences to the theatre? Step forward Jonathan Kent, departing artistic director of the Almeida. If his theatre were responsible only for staging the hippest, hottest show of the summer – twentysomethings flocked to Neil LaBute's The Shape of Things at the Almeida's new bus station home – that would be accomplishment enough. But to make Chekhov cool? Three years ago, Ivanov was among the Almeida's most irresistible hits; now, tickets for Platonov are shifting faster than furry hats in a Moscow winter. What have you done, Jonathan Kent, to render the bard of Russian boredom fashionable?

"For a start," he replies, "we've not just done Chekhov, we've done early, unknown Chekhov." Both Ivanov and Platonov are considered the playwright's "immature" works. Platonov was written when Chekhov was 20; in its original version ("an inchoate mess," says Kent), it sprawls over six hours. It's never really been shown in Britain. The Royal Court came closest in 1960, but cast as the play's 27-year-old hero, the 52-year-old Rex Harrison. "Considering that Chekhov is the greatest dramatist of the 19th-century," says his new adaptor, David Hare, "we should at least allow him to have this play seen, in English, in something like the form he intended." The buzz that surrounds the resulting, nipped-and-tucked version doesn't surprise Hare. "Just as my fantasy would be that Hitchcock had made five films in the 1950s which we'd never seen, people can't believe Platonov exists and they don't know it."

To Kent and Hare, the secret of the popularity of their Chekhov productions involves subverting stereotypes. "People are sickeningly familiar with the mature Chekhov," says Hare, "and with second-rate productions in which the plays seem very listless." According to Kent, "Chekhov seemed to represent a fin de siècle melancholy that we as a nation found attractive. But he's more volatile than that." With Platonov, the pair seek to bring out the playwright's youthful vigour – not a quality associated with Chekhov. "But there's this young writer who is much more direct, more political and more comic," says Hare. "He's furious about the state of Russia, he's writing overtly about sex, and he feels like our contemporary." Platonov is about "classic adolescent problems: how do you escape self-consciousness? Is it possible to be honest in a fallen world? If those questions aren't contemporary, I don't know what are."

If it's a play, then, with a natural appeal to modern audiences, the excitement that surrounds it is pure Almeida. The Islington-based theatre, run for 11 years by Kent and co-director Ian McDiarmid (they resigned from the job last week), has a genius for generating a sense of event. It has shrewdly tapped into the "found spaces" phenomenon: its current, post-industrial King's Cross home follows a season in Hitchcock's dilapidated film studios in London's East End. And it has led the field in "star casting": Kevin Spacey performed Eugene O'Neill here; Juliette Binoche, Ralph Fiennes and Anna Friel are also recent employees.

Platonov's creative team rebut the suggestion that the theatre's cachet is celebrity-driven: "[Platonov star] Aidan Gillen is not the best-known actor in England," says Hare, "yet as many people have booked to see this as booked to see Ralph Fiennes' Ivanov". Kent is defensive on this point, perhaps because the star-casting bubble is bursting. When Anna Friel fell ill several times during her run as Lulu (Martine McCutcheon wasn't the only one), audiences cried foul and the press let rip. But the director does agree that "theatre is going through a time of change where audiences need the adrenaline jolt of The Event. The old middle-class habit of going to the theatre four or five times a year is dead. And I don't know if I want to play to that audience, anyway."

To David Hare, the success of the Almeida brand – audiences now seem to trust the theatre, despite its occasional failures – is down to "the unfamiliarity of a European-style theatre" that explores an international repertory and programmes the likes of LaBute and Chekhov back to back, boosting the credibility of both. It helps that the Almeida's agenda – it sees theatre in primarily literary terms – concurs with that of several influential critics. The Almeida has performed a role for which some look, in vain, to the RSC and the National, at a time when we're sceptical of institutions and state-provided theatre. (The perception of the Almeida as an underfunded underdog may be intrinsic to its success.)

"We're living in a time of national nervous breakdown," Kent agrees, "whereby we've attacked every institution – royalty, the church, the police, the opera house – but we don't know how to replace them." It's a context that has allowed the Almeida to flourish. "This theatre is an act of the imagination, it's not an institution. The danger is when things become so institutionalised that no personality can shine through." Under Kent and McDiarmid's directorship, the Almeida has made theatre exciting by holding to the belief that, "a theatre has to be driven by the obsessions and visions of the people who run it. Ours is not a cynical exercise like, 'How can we create an event out of this?' We haven't patronised our audience," says Kent, "we've just done the plays that we love."

'Platonov': Almeida at King's Cross, London N1 (020 7359 4404), previewing, opens Tuesday to 10 November