Over the years the late-Fifties comedy double-act of Mike Nichols and Elaine May has acquired a legendary showbiz status. It was a casting coup, then, for the National to announce that Nichols - more famous for his career as a film director (The Graduate, Working Girl and, now, The Birdcage) - was going to appear in a new play by Wallace Shawn.

The playwright himself is better known as an offbeat film actor: he's the cuddly bald guy in My Dinner With Andre and Manhattan. His new play, The Designated Mourner, is not so accessible: a series of intercut monologues, it's witty, frank, and elegiac, on the one hand; obscure, confusing and draining on the other. In David Hare's production, however, Mike Nichols gives the most engagingly intimate performance in London.

The cast face us, in Bob Crowley's design, from behind a trestle table, set in front of a wall of gold-leaf tiles. There's bric-a-brac on the table: leather-bound books, jugs of water, flowers. It's like an arty press conference in a fantasy bank vault. Perhaps we're inside Nichols's mind. But then Nichols may be out of his mind.

Nichols plays Jack, a guy who can sum up his life in about 10 words: "a former student of English Literature who - who - who - went downhill from there". He now writes a sex column for a paper. When Nichols sets off on a self-deprecating riff, the other two other only half-listen. Miranda Richardson, his excitable, impersonal wife, Judy, fills her fountain pen or lights a cigarette, while David de Keyser, his scornfully genial literary father-in-law Howard, reads or stares balefully. They have their moments: Nichols has his quarter-hours.

We hear how he met Judy, how he hates Howard, how he left Judy, how he fell into despair. This picking away at scabs entertains us because of Nichols's enormous charm. When we discover he likes ice-cream, porn and suffers from a constant murmuring in the head ("idiotic arpeggios of self- approbation"), we like him all the more. Nichols stays seated: his narrow eyes glinting out of his squat, impish face. He looks as if he's constantly suppressing a burp. When he searches for a word, he waves a plaintive hand towards us or looks enquiringly off-stage. The performance way outstrips the material. I may never see Nichols doing stand-up, but after Designated Mourner, I feel I've seen him doing sit-down.

Shawn's play might be a cruel portrait of the anxieties that eat away at the middle-brow mind. Except that Shawn's own ambitions border on the Napoleonic. His minimalist play takes on the world. He derives his title from the tribal custom whereby, when the last member of a family dies, someone is designated the mourner. Fine, but which tribe exactly has become extinct?

Shawn suggests it's the cultured, wealthy elite. (Looking round the First Night audience, that particular tribe seems in good shape.) And who exactly are the ominous people alluded to as "dirt-eaters", who shoot literati in the back of the head during meal-times? It's frustratingly vague, the way Shawn sketches in these apocalyptic upheavals.

I hung in there, sustained by the irony and candour of Nichols's performance, as he struggles with increasingly embarrassing insights. But when Shawn broadens the focus - to suggest a moral and political vision of society - the picture blurs.

It's not often that a theatre programme carries an editorial from the Independent on Sunday. But Peter Whelan's new play at the Birmingham Rep tackles the timely issue of republicanism. Divine Right takes place in the year 2000, when Prince Charles decides he doesn't want to be King and his 18-year-old son has to decide whether he wants the job instead. To make up his mind, William gives his bodyguard the slip and spends three days in disguise meeting the rest of us.

Whelan, a republican himself, has the neat idea of making the young prince a clever, earnest, sensitive figure. So clever, earnest and sensitive that, after meeting a cross-section of British society - hooligan, shopkeeper, expatriate, whingeing parent and schoolteacher, he decides that what Britain needs isn't a monarchy. "Get a goat," he says, losing his temper in the newsagent's. "Take it up to a high cliff edge. Load all the nation's troubles on it, slit its gullet and fling it over!"

As the impeccable prince, the 18-year-old William Mannering delightfully catches the Royals' hesitancy and self-mocking humour. He is touchingly supported by his sympathetic younger brother (Christopher Trezise). There's a second, prosaic plotline about how the Labour Left and the Tory Right join forces to form a Republican party. This gives us, through standard House of Commons debates, the arguments for and against. It's the impact royalty has on people which is so well observed in Bill Alexander's stimulating production. Not even the female Labour Republican can stop herself curtseying when the prince arrives.

At the Almeida, Jonathan Kent assembles, for Moliere's Tartuffe, an enviable cast who deliver Richard Wilbur's very funny translation with rapid fervour. There's no sitting back pompously savouring the lines. This Tartuffe comes in at 100 minutes without an interval.

Rob Howell's striking design of a curving panelled corridor allows the verse and the actors to hold the foreground. They deserve it: Tom Hollander is a calculating Tartuffe, and Ian McDiarmid a sublimely blustering Orgon. Emma Chambers is hilarious as the daughter, Susannah Harker gorgeously composed as Orgon's wife, and Diane Bull delightful as the plain-speaking maid Dorine. It's a very funny, quick and surprising evening.

As the jury room fills up in Reginald Rose's Twelve Angry Men, we have the surreal experience of hearing British character actors - many well known from TV series - addressing each other in Brooklyn accents. Students of director Harold Pinter's own work will note the subject matter. A single juror persuades the 11 other members - in the face of ridicule and hostility - to change their votes. What looks like an open-and-shut case turns out to have many different interpretations.

Kevin Whateley plays the Henry Fonda role with a nice, unpriggish sense of just-not-knowing and just-wanting-to-ask. Timothy West is the crisply logical juror who takes the longest to convince. This powerful ensemble piece depends on the rich interaction of characters. This production has it. Any jury that includes the tolerant Alan MacNaughtan, the cadaverous Maurice Kaufman, the tentative Kevin Dignan and the staunch Douglas McFerran (to name only a handful) has enough flavour for a Bombay Mix.

Theatre details: Going Out, page 14.