Mark Rylance on Hollywood, the Oscars and why he won't be leaving the stage behind

He arrives for our interview wearing his trademark hat and keeps it on throughout.

Slight, gently spoken, and genuinely bemused it seems by the whole Hollywood blockbuster media circuit he now finds himself on.

Mark Rylance, after his stunning performance in Bridge of Spies, had better get used to this circuit of interviews and promotion.

The actor who tells me he went for many jobs in TV and film over the years and was turned down for all of them is about to become "in demand" - very "in demand."

He plays Rudolf Abel, a Russian spy in the true life story of a prisoner exchange that took place between America and the then Soviet Union in 1962, on a bridge spanning East and West Berlin - the so-called "Bridge of Spies."

It is a classic cold war drama, tense, dark, action-packed and Rylance, who plays opposite Hollywood veteran Tom Hanks, steals the show.

Not bad considering that when Steven Spielberg first came knocking on Rylance's door 30 years ago, when the actor was offered a role in the film Empire of the Sun, Rylance decided to choose the theatre instead.

Now Spielberg has his man. And not just in Bridge of Spies. Rylance has taken the lead role in Spielberg's remake of The BFG, the Big Friendly Giant based on Roald Dahls' much-loved story. That film's due out next year.

"How would you deal with an Oscar nomination?", I ask. He is hotly tipped to win the award next February. He grimaces.

The man regularly hailed as the greatest living stage actor of our generation finds the whole Oscars circus a bit daunting.

Rylance has won two Olivier Awards and three Tony Awards for his stage work.

He plays Shakespeare, Al Pacino famously said of him, as if it was written for him the night before.

On TV there was a BAFTA for his role in The Government Inspector. And although playing Thomas Cromwell in the BBC's acclaimed Wolf Hall, adapted from Hilary Mantel's novels, has brought him more recognition, he is hardly a household name. Yet.

He is right when he says that awards like the Oscars, the promotional tours etc, are all just a shop window.

But starring in a Spielberg film that has had universally glowing reviews, one senses that Rylance will be asked to play the Hollywood game a lot more.

The theatre will never lose him, he says - only if I die, he jokes - but how will he navigate a continued career dominating the West End and Broadway, when he has the draw of Hollywood films and a massive audience? It is an intriguing dilemma.

He has seen his theatre contemporaries like Daniel Day Lewis achieve huge success on film but says he has refused, in the past, from being made to feel like he has somehow failed or is not the complete actor because he has chosen theatre instead.

At the end of the press screening I attended for Bridge of Spies there was a round of applause - and not a few tears.

It is wonderful, one I will see again and Mark Rylance, star of stage AND screen, gives an award-winning performance.