If British theatre has had a golden era over the past dozen or so years you can pretty much put it down to the artistic directors of three of the country's most important producing houses: Nicholas Hytner at the National Theatre, Dominic Cooke at the Royal Court, and David Lan at the Young Vic.

Each in their own way has raised the bar for making theatre and running a theatre. All are Jewish but only one – Lan – is still in the job that in all probability represents the pinnacle of these stellar careers. Lan has held his role since 2000. One and a half decades later, the Young Vic was recognised at last Sunday's Olivier Awards as the powerhouse it is. The theatre had received 11 nominations, more than any other, and its production of Arthur Miller's A View From the Bridge starring Mark Strong bagged three awards, reaping the recognition that critics unanimously said it deserved.

The recent and current crop of Young Vic successes reflect the breadth of Lan's vision and his willingness to take a chance on the kind of theatre that, as the overused phrase has it, pushes back boundaries.

They include mesmerising revivals of Beckett's Happy Days starring Juliet Stevenson, and Tennessee Williams's A Streetcar Named Desire with Gillian Anderson's Blanche DuBois. Add to these the Miller, and you could be forgiven for assuming that Lan leans heavily on the tried and tested. But the opposite is true.

Streetcar and Bridge especially show a taste for not just producing famous plays but staging them in ways that strip away the musty conventions that's often associated with them. The effect has been profound. For those who have never seen the plays it feels as if they are watching something that has been freshly written, and those who know them well are forced to view them as if for the first time. Benedict Andrew's production of The Three Sisters was Chekhov for the grunge generation. Where the play calls on Russian folk music to be played, the audience were instead thrillingly regaled with Nirvana's Smells Like Teen Spirit. You could almost hear the pop of monocles falling out of the eyes of traditionalists.

It's a strategy that Lan developed alongside the more obvious method of ensuring healthy box office – casting A-list stars. Jude Law in an intimate Doctor Faustus set the tone.

But what sets the Young Vic apart is Lan's outward-looking vision. It is not so much the stars on his stage that has vaulted the theatre's reputation beyond that of even the (better publicly funded) National, but his willingness to pull in directors from abroad. It is this that has made the Belgian Ivo van Hove – the visionary behind the reinvention of A View From the Bridge – man of mode among directors. And that production of The Three Sisters was directed by Benedict Andrews, an Australian, as was Streetcar which was set in a modern apartment that seemed to be furnished by Ikea.

This internationalism is no small thing in a country whose theatre establishment is capable of looking at European theatre-makers with the scepticism most of us reserve for the Eurovision Song Contest. But Lan's horizons are broader and reach further than most of his contemporaries. This might, in part be down to his roots. The grandson of Lithuanian immigrants, he was born in South Africa. It was, he once told this newspaper, "a classic Jewish experience".

His father's parents came from a ghetto called Slabodka. When the family attempted to move to Russia in 1914 they became embroiled in the turmoil of war. His great grandparents, who died of typhus, were photographers and Lan still has some of the pictures they took of the shtetl's children. It's not too fanciful to suppose that they were at least in part the inspiration behind the National Theatre's shtetl-set play Travelling Light, which was written by Nicholas Wright with whom Lan lives. By the time Lan's grandparents Golda and Mottel had made it out of the Soviet Union to South Africa the family "were Trotskyites", though that didn't prevent them from going into business in the "coloured" suburb of Cape Town where they ran a shop selling bicycles and records. Their grandson David first came to London in 1970.

It is, perhaps, just a little bit amazing that the theatre which feels like the youngest and most vibrant in the capital has now been run by the 62-year-old Lan for 15 years.

These days devotion is split between the Young Vic and helping to set up a new arts centre planned to open in the World Trade Centre in 2019. The visits to New York are week-long and monthly.

But that should be no problem for a man with a healthy disrespect for the boundaries and borders that apply to countries as much as it does to theatre.