THERE WAS a rumour that Stephen Daldry was going to direct Fiona Shaw in Mary Poppins. Later the National announced that Deborah Warner was going to direct Fiona Shaw in Private Lives. Neither happened. Then, last week, she opened at the National in The Prime of Miss Jean Brodie. She's found a role that combines the running-around-with-children bit of the first part with the sophisticated hauteur of the second.

Fiona Shaw makes a startling entrance, spot-lit at the back of a cavernous Lyttelton stage, in a red outfit. Funnily enough, the last person to make a startling entrance on to the Lyttelton stage wearing bright red was Maggie Smith in Stephen Poliakoff's Coming in to Land. But we won't mention her name again in this context.

Shaw walks downstage with the showbizzy self-regard of a gameshow hostess. As a woman in her prime, whose pupils are the creme de la creme, Brodie has just returned to 1930s Edinburgh from her summer holidays in Italy - she's a fan of Mussolini - to teach her girls the value of Goodness, Truth and Beauty. They are a precocious bunch. In Phyllida Lloyd's spectacular opening sequence, they greet her with a tableau vivant of the Last Supper. Appropriately - as one of her girls will betray her.

Shaw's Brodie manages to be nervy and statuesque. Everything is on a precarious tilt. Her hair sweeps back. Her head leans sideways. One foot turns on its side. She strikes heroic attitudes. She plays someone giving a performance. Her palms are held up high. One hand sits on her hip. When she practises her golf strokes she wiggles her behind as if trying to swat a wasp. As we watch her throw out Muriel Spark's epigrams in her Cork accent she seems a better choice to play Oscar Wilde than Liam Neeson. She has the vitality and intelligence. She has the mind.

Of course she's head and shoulders above the girls. Literally. They act as her chorus, her backing group, her corps de ballet. A ripple effect takes place: the girls become an extension of her personality. Within 10 minutes, it's apparent what Phyllida Lloyd's direction and the newly revised adaptation by Jay Presson Allen have done to Muriel Spark's novel. They are opening it out, finding scenes in the scene changes, adding songs and orchestrations, taking it halfway towards becoming a musical. There are neatly choreographed sequences with slide projections as Shaw takes the girls round Edinburgh. The depth of the stage is used to great effect. Miss Brodie has a scale we hadn't expected. Just as we hadn't expected it in Trevor Nunn's pro-duction of An Enemy of the People.

Nor in his production of Tennessee Williams's Not About Nightingales. It's to do with music, movement, fluidity and pizazz. And big budgets. It's to do with Cats and Les Mis too. It's the Trevving of the National.

David Mamet's The Old Neighborhood, which premiered last year in the States, and now opens in London, is a fascinating trio of plays. In The Disappearance of the Jews, Bobby talks to an old friend in a hotel room. In Jolly, he talks to his sister and her husband in their kitchen. In Deeny, he talks to a woman in a restaurant. Bobby has walked out on his wife and kids. His suitcase sits on the stage. This is his journey.

Mamet opens with a bravura piece of ruminative stock-taking - seemingly casual and random. Colin Stinton's Bobby and Linal Haft's Joey sit drinking in a hotel bedroom talking of their childhood, girls they knew, broads they knew, and shtetls, shiksas and schmucks. They struggle with a sense of incompleteness. Haft is hilariously emphatic as Joey frets about masculinity, marriage and Jewishness. In a terrific performance, Stinton catches a pained, slow-burning intelligence. Self-contained, he sips his Dr Pepper and drums his fingers. Intentness spreads across his face like stubble.

As his sister, Jolly, Zoe Wanamaker fills her kitchen with indignation as she flays her parents for their chilly disapproval. It's a text-book piece of "and another thing" acting. One grievance sparks off the next. Wanamaker's feistiness has such warmth that it never seems a whinge. With her doe eyes and ski-jump nose, her features give her severity a comic edge. With these performances Wanamaker and Stinton put themselves straight into the awards categories.

Deeny is the least successful play. An old girlfriend of Bobby's (Diana Quick) is now divorced. As Quick and Stinton sit on the restaurant banquette she talks her way out of reigniting the relationship. Everything we learn in the first two plays adds to our knowledge of Bobby. Many layers of his life have been hinted at and sketched in, but here we lose the focus.

The dialogue in The Old Neighborhood pulses with an electric charge. Characters keep rewriting what they say. Thoughts stack up. Replies are delayed. Mamet peppers his dialogue with "heys", "huhs" and "uh-huhs", placed as precisely as musical notes. Naturalism like this is a question of enormous artifice. But here it seems effortlessly natural.

This London production drives the action forward, knocking 20 minutes off the New York one. The designer William Dudley introduces a revolve which speeds us through scene changes. With this powerful production Patrick Marber threatens to woo away punters from his own Closer. The two productions prove that - as a director as well as a writer - he's a sharp stylist with a firm grasp of rhythm and form. He could do Pinter or Coward. Private Lives, maybe.

Paula Vogel's How I Learned to Drive opened off-Broadway last year and won a Pulitzer. The subject is that old News at Ten standby: paedophilia. Set in the rural backwaters of Maryland in the Sixties, Uncle Peck (Kevin Whately) becomes sexually obsessed with his niece, Li'l Bit (Helen McCrory). He teaches her to drive and, while doing so, tries to molest her. This is told from the girl's point-of-view. At 18, she rejects him for good. Seven years later he dies, an alcoholic.

What's amazing about this play is its mildness. Li'l Bit is sexually precocious. Uncle Peck doesn't force himself on her. He never rapes her. They never even have sex. If they had had sex, it wouldn't be incest as they're not blood relatives. If this is the extent of what was going on in the backwaters of rural Maryland in the Sixties then Marylanders have plenty to celebrate. It's a hundred times better than what's been going on in children's homes in Britain.

This isn't a tragedy, it's a sad, pathetic tale. But in John Crowley's production it is treated with queasy laboriousness. We hear about each step in the relationship as we watch Whately encourage Li'l Bit to drink Martinis, or pose for modelling photos in his basement or take hold of the steering wheel.

The cast performs this case-history, an overblown anecdote, in a jokey, presentational style: the show might be en route to the Edinburgh Fringe. The narrative jumbles up the chronology. Each scene is preceded with an arch quotation from a driving manual. Other members of the family are broadly conjured, with the young Michael Colgan and Philippa Stanton (two years out of drama school) playing grandparents. The strength of the evening lies in McCrory's commanding performance. But even that is odd: an undeniably sexy 29-year-old woman is playing a pre-teen and teenage girl. You couldn't blame Whately for looking troubled.

'Miss Jean Brodie': Lyttelton, SE1 (0171 452 3000), to 3 Oct. 'Old Neighborhood': Royal Court, WC2 (0171 565 5000), to 8 Aug. 'How I Learned to Drive': Donmar, WC2 (0171 369 1732), to

8 Aug.