Those of us growing up in the heady days of 1960s Liverpool knew that four local lads were taking the world by storm. Some really grown-up people might even have been to The Cavern and seen the phenomenon in their early days. And yet there was always an enigma in the background: the figure who made it happen but about whom we knew almost nothing.

Something new is happening in the West End. Just up the road from Thriller and down a bit from Les Misérables a billboard the colour of weak tea (positively consumptive compared to the full-colour, neon assaults on either side) proclaims the arrival of Richard III and Twelfth Night. Shakespeare is back on Shaftesbury Avenue, and this time he means business – big, commercial business. How has this sleight of hand been achieved?

1866 was a crucial watershed in Henrik Ibsen’s writing career. As a man he may have come of age some 20 years earlier, but it was only at almost 40 that his writing attained brooding, bearded maturity in Brand, the first in the sequence of plays that we now accept as the Ibsen canon.

Nick Payne has already made quite a mark. In 2009 he won the George Devine award for Most Promising Playwright with the intriguingly entitled If There Is I Haven’t Found it Yet at the Bush. Wanderlust followed at the Royal Court and now with his second Court commission, transferred to the Duke of York's from Upstairs at the Royal Court, he’s come up with bees and multi-universe theories, love and death.

There’s no attempt to romanticise the hero of Nick Dear’s new play about the Anglo-Welsh poet Edward Thomas. Thomas, who died in action in the Battle of Arras in 1917 after enlisting at the age of 39 – far too old to have had to fight – is played by Pip Carter as prickly, petulant and with an alarmingly misogynistic streak. He tramples over the feelings of his adoring wife Helen and displays an unattractive physical cowardice when ambushed by an angry gamekeeper on one of his long country rambles. 

Science thrives on stage. In play after play, various scientific ideas seem to flourish in the warm, well-lit environment of the theatre, fed by a crew of artists and despite the threats of critics or other predators. Now, Lucy Prebble — fresh from her outstanding success with Enron — turns her attention to the subject of love and neurology in her latest play, which opened last night. Directed by Enron maestro Rupert Goold, the play stars Billie Piper so it’s already sold out, but is it any good?

Not even a cameo by Tamsin Greig can redeem this painful adaptation of Euripides' The Trojan Women. For an hour and a half it screams with anguish, verging at times on the parodic. The production is a puzzle. Caroline Bird has updated the language, stripping the original of much of its poetry and adding expletives.

This is a short play, but not a sweet one. Nevertheless, the ban on under-16s and the warning that it “contains themes that some audience members may find distressing” seems unnecessary for more than 50 of its 70 minutes.

Howard Barker is hardly known for light entertainment. In The Europeans, a raped woman gives birth on stage. In Scenes from an Execution, currently at the National, a Renaissance artist is at war with her patrons. In Lot and His God, based on the Genesis story set in the wicked city of Sodom, Lot's wife betrays her husband with an angel. Complex might be putting it nicely.

The word “people” of the title of Alan Bennett’s new play is to be spat out, like a lemon pip. People, who invade your space, boss your values, make you be what they want. So does the beleaguered Lady Dorothy Stacpoole feel about the stark options facing her as her fantastically grand mansion leaks and crumbles over her smelly, freezing feet, while under it groans ancient mine workings like a whale with toothache.