Polar Bears, Donmar Warehouse, London<br/>Andersen's English, Hampstead Theatre, London<br/>Beyond the Horizon/Spring Storm, NT Cottesloe

Kay is definitely not OK. We learn she is a manic depressive, then we hear she's lying murdered in her basement.

But I'm telling it wrong. Those events are, in fact, relayed in reverse in Polar Bears. Starring Jodhi May as Kay, this is a structurally intriguing debut play about those who care for and/or harm the mentally ill. It's written by Mark Haddon, hitherto best known for kids' stories and The Curious Incident of the Dog in the Night-Time, his novel narrated by an autistic boy.

If told straight, Polar Bears' storyline would run as follows. Kay's father commits suicide. Her brother (Paul Hilton) turns into a little bully, while her mother (Celia Imrie) says her daughter should never fly the nest. Kay does go to uni, but becomes a needy fabulist, prone to bipolar mood swings and visions of Jesus. Her real-life saviour is Richard Coyle's loving John, a gentle philosopher. Increasingly stressed, though, he turns psychotic himself.

Haddon does not simply move backwards, but more craftily disorders the plot, jumping to and fro in time. Thus, on the one hand, we're gradually piecing together the sequence of factors contributing to Kay's grim fate. On the other hand, because of the scrambling, we're experiencing confusion, as if we're seeing things replayed or unreliably dreamed up in an unstable mind – Kay's or John's.

Ordinary situations mesh with hallucinations, sometimes comically, sometimes menacingly, on a peeling grey set – like some mental dungeon. And you keep wondering about Kay and her loved ones even after you've left the theatre. Have all these people, to some degree, been driven mad? Are they each deluded about Kay in different ways? Are we even sure she's dead?

What's disappointing is that, while the narrative structuring is clever, May fails to make Kay poignant. When she's meant to be hypermanic, Haddon has her launch into pseudo-poetic spiels that sound overwritten. Like her creator, she's also a kidult author, telling an interminable fairy tale about a princess whose bosom buddy is a monster. Oh, spare us.

Better, Coyle ranges from the charmingly witty to the dangerously febrile. And Hilton switches between satirised git and unexpectedly affectionate sibling. The evening is peppered with strong and startling dramatic moments: a commendable production and promising first play, but still a bumpy ride. Kay isn't the only one suffering ups and downs.

Hans Christian Andersen is deeply eccentric, if not completely cuckoo, in Sebastian Barry's new biodrama, Andersen's English, about two privately dysfunctional literary geniuses. In Out of Joint's touring production, directed by Max Stafford-Clark, Danny Sapani plays the Danish children's author fetching up without warning at Charles Dickens's family home in 1857. His luggage includes a long rope, in case he had to escape from an upper storey, but then he outstays his welcome. Socially blundering, he veers between latently homosexual adulation for his host and moaning rudeness towards others.

David Rintoul's patriarchal Dickens remains polite to his house guest while nastily pushing his ageing wife, Catherine, out of his life. He sends their dallying son, Walter, packing too, but amorously retains Catherine's housekeeper-sister.

Interesting biographical material. Awful dramatisation. Andersen's English rambles on, stuffed with speeches that re-create Victorian floweriness but never sound natural. Lucy Osborne's set caters for indoor and outdoor settings with magnified floral wallpaper, but puppets playing additional offspring don't liven things up much. Nor does Andersen's tediously unfunny, lousy English. Poor Sapani just resorts to sing-song.

The main interest lies in giving a voice to the historically sidelined Catherine, played by a deeply galled Niamh Cusack. Rintoul, meanwhile, only scrapes the psychological surface of Dickens, rattling through his lines, though one sympathises with the actor's desire to see the back of this whole protracted episode.

Lastly, a brace of early works by Eugene O'Neill and Tennessee Williams – transferring to the National from Northampton's Theatre Royal – gets off to a cracking start. O'Neill's Beyond the Horizon (1920) is a remarkably assured, decade-spanning saga in which two brothers on a Connecticut farm tragically ruin their prospects when they try to resolve a love triangle. The bookish Robert (newcomer Michael Malarkey), suppressing his wanderlust, stays home in a souring marriage, while the born man of the soil, Andrew (Michael Thomson), heads for the high seas. Laurie Sansom's production offers beautifully detailed tensions and yearnings, with particularly fine work from Liz White as the eager girl-going-on-bitter wife. Malarkey increasingly tends towards blank-faced gloom, however, so his terminal tubercular fever isn't gripping.

Tennessee Williams's Spring Storm was written in 1937, when he was studying playwrighting and reading Beyond the Horizon. The two pieces are consequently full of enriching parallels, and Sansom creates fluid twin productions, with doubled actors and stylised scene changes which foreshadow the way the young romances twice end in wreckage.

It is, nevertheless, avaricious snobbery that drives Williams's small-town Southern belle, Heavenly Critchfield (White), to ditch her brawny sweetheart, Dick (Thomson), in favour of the screwed-up college boy, Arthur (Malarkey). Some of the characters lack psychological consistency, but Anna Tolputt shines as the shy, ardent librarian, devastatingly left on the shelf by Arthur. Moreover, the older generation is splendidly satirised with Jacqueline King as White's neurotically uppity mother and James Jordan rolling his eyes as her laid-back dad. Enjoyable.

'Polar Bears' (0844 871 7624) to 22 May; 'Andersen's English' (020-7722 9301) to 3 May; 'Beyond the Horizon'/'Spring Storm' (020-7452 3000) in rep to 19 Jun

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