THERE is an appealing, elfin quality about Mark Rylance, the artistic director of Shakespeare's Globe. At 39, small and twinkly-eyed - or, to quote the producer Thelma Holt, a friend and admirer, "a little runt of a thing" - he exudes a boyish earnestness which is, of course, only part of the story. He takes himself a mite less seriously than people assume and is not in the least bit grand.

On the day of announcing the Globe's new season last week, Rylance coped courteously with hours of interviews, questions and photographs. The building may be more important than any one person but, for many, he is synonymous with the Globe.

His chatty openness has led him into difficulties in the past. The adjectives "hippy-dippy" and "loony" used to be permanently attached to his name. And it is true that his head is full of ideas that he espouses with fervour. He describes himself as an amateur philosopher, fascinated by the Egyptian sage Hermes Trismegistus, just as Renaissance dabblers in magic were. He admires Plato but also describes himself as a pantheist. He discusses the power of landscape, the energy of matter, the division of a human being into three - the physical, the soul and the spirit - and then insists, "But I'm very practical, I'm grounded."

His most surprising decision for his third full artistic season has already caused a furore: he has cast himself as Cleopatra in an all-male production of Antony and Cleopatra. Why? Shakespeare is full of challenging male roles; Rylance does not need to prove himself as an actor - his 1989 Hamlet was for many a definitive interpretation; and his Ulster-accented Benedick in a West End Much Ado brought in thousands unaccustomed to Shakespeare.

His answer might not satisfy all the female would-be Cleopatras (although, to be fair, he claims to have received no complaints from actresses) but he presents it thoughtfully. "In the first season, audiences enjoyed the creative role they played in Toby's success. [Toby Cockerell played Princess Katherine in an authentically all-male Henry V.] They lent him their imagination. It wasn't all done for them."

We've become literal-minded, he says: "Film actors have to prove they can really box or live with disability, whereas playing something you're not is what makes the most of the imagination. Besides, Shakespeare was exploring the marriage of opposites, of feminine and masculine, within characters."

Rylance was deeply impressed by the all-male Kabuki theatre he saw in Japan last autumn when the Globe's As You Like It was on tour there. He enjoys finding connections between things, and was pleased to learn that Kabuki began in 1603, a few years after the establishment of the first Globe on Bankside. He also likes the idea that at 39 he is within a year of Cleopatra's age at her death.

Casting Antony is proving a bit of a problem, though. Are actors embarrassed at the prospect? "Oh," he laughs, "I hadn't thought of that. Perhaps they're too polite to say so."

He cites Kathryn Hunter as an example of an actress who refused to be limited by her gender and played Lear. She will direct The Comedy of Errors as another part of this Roman season, which also features Julius Caesar and a new play about Celtic Britain, Augustine's Oak, by writer-in-residence Peter Oswald. Rylance still plans an all-female production some time "but it would be too politically correct this year." Clearly, he doesn't believe in making things easy for himself.

He has had to show resilience in the past, especially when, just after his appointment to the Globe in 1995, his Greenwich Theatre Macbeth, set in an Eastern cult, was savaged by the critics. He was deeply shocked by the "vehemence and violence of the response to an experiment - as if we had made decisions without deep consideration." He laughs again: "My imaginative world differs from the norm - or so I discovered then."

This summer Rylance is to direct for the first time since then - a piece that in 1999 has special significance. Julius Caesar was the first play performed at the original Globe, a building constructed from timbers smuggled by the cash-strapped Burbages across the Thames, beyond the jurisdiction of the City, from the dismantled theatre in Shoreditch. This was when the opportunity arose for actors, including Will Shakespeare, to buy shares in the company.

Asked if he is putting himself under unnecessary pressure, Rylance says that he has reached a stage when he wants to give more to the project. Last summer Peter Kyle arrived as general manager, which has freed Rylance from some administrative responsibility.

Still short of money to develop both education centre and exhibition space, the Globe receives no state subsidy. He and his wife, the musician Claire van Kampen, with whom he says he shares all decisions, have devised a collaborative system in which the "Master of the play" will work closely with a "Master of verse", thus sharing the directing workload.

Two plays with elements of tragedy, then, a daring repertoire for a space that encourages uninhibited audience response, but Rylance reckons they are ready. During the first season people fainted in unexplained numbers: "Some nights it was like ER here without George Clooney." Presenting anything too emotional might seem to be asking for trouble. But it hasn't happened since and after a season that was "all comedies" it seemed time to have a go at something heavier.

He clearly enjoys the freedom of his position and candidly admits, "Of course I can choose any part I like." His latest enthusiasm is for the poet Murray Lachlan Young (he of the reported £1 million contract with EMI) who wrote a poem about the historic "flit" across the Thames on its anniversary in December. It seems likely that he will provide something suitable to be declaimed before performances of Julius Caesar and he is contributing a new sonnet for Shakespeare's birthday celebrations.

Rylance loves mysteries, and that has led him to question the authorship of Shakespeare's opus. He favours Francis Bacon because "there isn't a single legal mistake and because of the vocabulary, the knowledge of Latin, Greek and French." He is cautious on this subject, though, having experienced mockery in the past, and would prefer to be seen as an explorer, someone who relishes all possibilities.

For all his openness, he remains something of an enigma. He is still fascinated by stories - that is what the Globe is about, he says - but there is no shortage of serious purpose to put them to. This year's theme is good government, and he is collaborating with Charter 88 to set up a debate at Middle Temple Hall in April. And no doubt he will manage to make complicated connections between Romans, Elizabethans and those of the Third Way. The end of the Millennium is, he says, a suitable time to consider these things.

The Globe's popularity is phenomenal; last season the company played to 89 per cent capacity, 98 per cent in the case of The Merchant of Venice. It is on every London tourist's list but it is also rooted in the local community, with thousands of Southwark children claiming it as "their" theatre. People who have never previously visited a theatre, who don't know how the Shakespearian warhorses end, stay to be swept up in the informal atmosphere.

No doubt the film Shakespeare in Love, in which an audience is moved, for the first time ever, by the last scene of Romeo and Juliet, will send even more visitors in search of an experience that they hope will transport them, if not to Elizabethan England, at least to a place where the spirit of Gwyneth Paltrow might be discovered supporting the case for cross-dressing.