Dominic Cooke is about to leave the Royal Court, where he has been artistic director for seven years. When, on 12 April, he steps through the door into London's Sloane Square, it will mark the end of a remarkable chapter in the theatre's history. The Royal Court has prided itself on being a home for new writing from the beginning, when its founder, George Devine, declared: "Ours is not to be a producer's theatre, nor an actor's theatre… but a writer's theatre." It was Devine in 1956 who put on John Osborne's Look Back in Anger, which redefined, in its portrait of seething, working-class Jimmy Porter, what a play could be and what a theatre could do.

Since then, the idea of the "writer's theatre" has held fast with each artistic director in turn. Max Stafford-Clark's school of playwrights was especially starry, including David Hare, Howard Brenton and Caryl Churchill. Ian Rickson had an eye for new writing too. It was Rickson who first put Jez Butterworth's work on stage and who directed the anarchically magnificent Jerusalem, a show that had people sleeping outside the theatre for returns. It ran during Cooke's regime and became one of the jewels in his crown.

There have been other ground-breaking box-office hits too: Lucy Prebble's Enron, Laura Wade's Posh Bruce Norris's Clybourne Park and several transfers to the West End and Broadway. But what makes Cooke's reign unique is that he has used the Royal Court's young writers programme as a way of finding and cultivating new talent, often by precariously young writers. Under Rickson and his predecessors, the programme had been about education. First plays were seldom performed.

But for Cooke, if a play was good enough, that was enough: he would put it on. The young writers programme was a bit like a kitchen from which good food could be served, piping hot, at the table next door. Polly Stenham's That Face, staged when she was only 19, bowled over its audiences. Anya Reiss was younger still – 18 –when her assured debut Spur of the Moment opened. Bola Agbaje (a comparatively ancient 25) won an Olivier award with her first play Gone Too Far!.

There are too many playwrights to squeeze comfortably into a single photograph but we have invited 11 – a dramaturg's dozen – to come and say goodbye. And it turns out that every one of them, with the exception of Bruce Norris (who is from Chicago), attended the Royal Court's young writers programme or has had work showcased in the young writers festival. Stenham is not alone in volunteering she would probably never have known she was a playwright without the Royal Court. Cooke's class turns out to be quite a crowd.

Bruce Norris's The Low Road is to be Cooke's last production as artistic director. I catch up with him in a rehearsal break. What is most striking is his receptivity – the quality of his attention. I like him at once. But there is something almost laughably traditional about his appearance – the V-necked sweater, the office shirt – for someone with such an eye for the new. He describes himself as driven by the need to "stay ahead of the game – and find work that is fresh, because that is the theatre's USP". At his first press conference, he famously promised to "explore the fraught paradoxes of what it is to hold liberal views … what it means to be middle class, what it means to have privilege and power".

He comes from a middle-class background – his father was a film editor, his mother an NHS receptionist. Now, he says, too much was made of this fleeting comment (the middle classes like nothing better than to talk about the middle classes). He wanted a shift in emphasis – away from "victims of the system". He believes "questions of responsibility make good drama".

Laura Wade's Posh was a better-than-good drama about an Oxbridge-educated upper class, members of the Bullingdon club, who believe prospering is their birthright. It opened – with canny programming – in the runup to the last election and then transferred to the West End as David Cameron, ex-Bullingdon himself, came to power. Wade believes Cooke's instinct about the middle classes and "untapped drama" was spot on. What was needed was a "new rigour" that would not be "cosy".

Yet it is important to register, as Royal Court playwright DC Moore points out, that the range of programming has been broad: "If theatre is going to earn its subsidy, it has to be for everyone. People talk about 'well-to-do' work in Dominic's regime instead of the work of Debbie Tucker Green, Hayley Squires [or] Anthony Neilson, which speaks of how most of us actually live or see the world." Not to mention DC Moore's own work, plays such as The Empire, written with subtle, political force. "If Dominic was only putting on middle-class plays, I wouldn't have had any on at the Court," he says.

The Observer's theatre critic, Susannah Clapp, refines the point, saying if Cooke had limited himself to "power and privilege", it might have been a "fairly female-free zone". Instead, he has got us to the point where "journalists no longer feel it necessary to call dramatists without penises women playwrights".

From left: Laura Wade, Mike Bartlett, Bruce Norris and DC Moore at the Royal Court. Photograph: Suki Dhanda for the Observer

More than 130 plays have been staged during Cooke's regime, which began in 2006; 94 have been full productions of new plays, the rest have been public readings and productions of old plays. A cool look at the programming shows how eclectic Cooke is and secure about his choices. He survived the risk of being thought nepotistic when he programmed his partner Alexi Kaye Campbell's play The Pride. And he has not turned his back on the old school either – there was his massively successful revival of Arnold Wesker's Chicken Soup. There have also been the flops – the Wallace Shawn season described by one playwright as "bonkers". EV Crowe puts her finger on it when she says: "I don't feel I can predict the boundaries of his interest or of his taste."

In January 2010, in an interview for this newspaper, Cooke had positive things to say about the Conservatives – Jeremy Hunt, then culture secretary, had been to see Enron and Cooke believed they were "quite interested" in theatre and "making all the right noises". He added: "What has improved is that there is now an acknowledgement that culture matters, economically and in terms of who we are as a nation."

How does he feel now about what he said then? "I am gobsmacked by what has happened. I was mistaken … I was told categorically by Jeremy Hunt, 'We will not be cutting funding.'" Hunt would later tell him: "We are in a different world now." Cooke will be leaving the theatre in good financial shape, but is concerned that politicians are "not getting it, not understanding how central the arts are to the financial infrastructure of the country, to its value system, to its soul". He describes government arts policy as "incredibly foolish".

And he is leaving at a critical moment. His successor, Vicky Featherstone, will be taking over at a point where theatre is said to be "shrivelling", according to the latest research based on a survey of 26 theatres by playwright Fin Kennedy and Helen Campbell Pickford, a doctoral candidate at Oxford University. They say "research and development" are suffering most. A 30% cut in Arts Council funding means that workshops, readings and writing schemes are being severely pruned. To the casual reader, this might seem no more than mildly troubling, but if one looks at the Royal Court as a model, one can see these findings are devastating.

The Royal Court's young writers programme is a reminder that writers need full-time commitment. The theatre receives about 3,000 unsolicited scripts a year, but it is rare for someone to have a play put on by taking this route. The most successful plays are collaborations that take place over a period of time. They grow out of conversations, commissions, workshops, readings and writers' groups. The Royal Court has always understood that writers are born and made.

Stenham says of the young writers programme: "It changed my life, didn't it?" But it is delicate. As Cooke makes clear, there is a "huge responsibility" involved in shaping young people's careers – in changing their lives. He remembers that with Anya Reiss: "We had to delay production because of her A-levels – that was quite an intervention." The theatre is also aware that you can't make a name one day and then turn your back the next. Ongoing support for young writers is essential: "We want them to feel part of what we are doing, to keep reading their plays. We want them to see we are there for the long haul."

The young writers programme is run by the playwright Leo Butler and open to anyone aged between 18 and 25. Unsurprisingly, it is oversubscribed (there are 100 applicants for 15-25 places). You have to submit a script and are judged on merit. It runs all year round with workshops, lectures and meetings with playwrights. What might surprise the sceptical is just how much it is possible to learn about technique.

Nick Payne recalls an exercise called "Open Space, Open Time", which explored ways of using time and space ("closed" equals real time). "Suddenly you know what the decisions are," he says. "What period of time is my play set over? How many locations does it span?" It was the perfect warm-up for Payne, whose West End hit, Constellations, took brilliantly considered liberties with time and space. The playwrights also talk about the motivation of being in a working theatre, the sense that the stage – and a possible future – is a stone's throw away. Being in the company of your peers is important too. EV Crowe explains: "It makes you feel part of a generation that matters, so you are not always looking to the older generation for how to do it but can back yourself and your own ideas."

Cooke worked at the Royal Court for four years under Ian Rickson before running it (before that, he was at the Royal Shakespeare Company for four years as assistant director). Nina Raine, who has worked with him as a writer and as a director, says: "I am in awe of Dominic's ability to see the potential in a play. Sometimes I'd privately think, 'God, why does he want to direct that?' But once he had brought it to life, you'd see …"

So what is he looking for? "I am always looking for something you have not seen before," says Cooke, "an idea, an argument, a formal concern, a voice … a play that takes something familiar in a new or unexpected direction."

This often means taking on work that is not finished, being able to identify potential, guess at a destination before it has been reached. He also has a gift for fine – and sometimes not so fine – tuning. Nina Raine describes getting "big, tough notes" and Laura Wade goes further. They are "killer" notes, she says appreciatively.

What does he ask himself when he watches a run-through? "I'll ask: is there a dynamic at the centre that can sustain the play?" The dynamic is everything. Cooke is mindful that writers work "through the unconscious" and aims to "ask questions that lead to a place where they recognise what they are doing and can make it more complete". Dramaturgy is a "fine art" in which he is supported by a team of directors, playwrights and a literary manager. He sees it as important to give writers the confidence to leave their comfort zones: Anya Reiss is working on a play that will be a new departure; Polly Stenham's No Quarter has had its original structure overturned; Nina Raine remembers how a question about the deaf boy at the heart of her play Tribes helped her put anger into words. It was a "big steer" – in the right direction.

Cooke has no idea about his own direction. "People are very suspicious," he laughs. "They think I know what I'm doing next." He needs a break and, in his case, it may actually be gardening leave, as he mentions that he is a keen gardener. Having said that, at 47 and with his track record, he is unlikely to be lingering in the flower beds.

What emerges is that the job of running the Royal Court, although exhilarating, has not been easy. He is a perfectionist. Responsibility is a word that keeps coming up. He finds management challenging and claims to be no good at it. He does not enjoy the "sensitivities" involved in rejecting work. And the hours are long. In his first year, he would work from 8am until 11.30pm without a break. Not that he had been intending to grumble. Working with actors and writers has, he volunteers, been his greatest pleasure.

And as a middle-class person, what has he learned about the plays about the middle classes? The greatest revelation he mentions is that "ownership" is at the heart of what it means to be middle class – and that property can be a sedative. About these matters, Bruce Norris has been his guide. The first play of Cooke's regime was Norris's The Pain and the Itch. The Low Road is his last. "I am a human bookend," Norris says with a laugh. Cooke does not want to say too much about their finale, but he does let slip one remark that, given the six years behind him, gives you pause for thought: "It is the most ambitious play I have ever worked on."

Polly Stenham

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

"The Court feels like home. I have grown up here – I think some of the big moments in my life happened in the theatre's bar." Polly Stenham is 26 years old now, dressed from head to toe in feline black. Her plays about dysfunctional, middle-class families have had an innate charisma that has won her many fans. She was just 19 when her first play, That Face, was hailed by Charles Spencer in the Daily Telegraph as "one of the most astonishing debuts I have seen in more than 30 years of theatre reviewing", while a smitten Michael Billington in the Guardian compared her to Noël Coward. And, at 21, she was the youngest playwright since Christopher Hampton to have a play transfer to the West End. But it is not uncomplicated to be successful at 19.

No wonder that when her second play, Tusk Tusk – about the toxicity of mental illness in privileged families – was on, she was "completely stressed out" and remembers "the theatre manager putting me in a dressing room to sleep. They never patronised me – if I didn't get in touch for a bit, they would call, make sure I was OK. They take you on as a person." They also "see the potential in you – you are a long-term bet". She adds: "I have always written for Dominic in a way – have wanted him to be proud. I am really sad he is leaving … because you go as well. That's the tradition. You go and then come back when you are a bit more established. It is the right time, but it will be emotional."

DC Moore

From left: Laura Wade, Mike Bartlett, Bruce Norris (in front), DC Moore (Dave). Photograph: Suke Dhanda for the Observer

DC (David) Moore, 32, had his first play, Alaska, on at the Theatre Upstairs at the Royal Court in 2007. "It was like a fairy tale because in the previous regime my play had got lost in the cracks and, without wishing to be rude about it, one of the most difficult things in this industry is actually not getting a no." Development in other theatres had become "an almost film-like process where people were doing it almost for the sake of it". The Royal Court was different. When he heard his play had been accepted, he had "20 seconds of joy and then absolute fear … I left the building and thought, I've got to write a play for Upstairs … Jesus, that's difficult." And it was not helpful that Stenham's play had just been hailed the best debut in a quarter of a century. "A funny thing to happen before your first play goes on." But Moore, an engaging character with tremendous eloquence, has gone from strength to strength, revealing in the well-received The Empire a pitch-perfect ear for dialogue.

Nina Raine

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

Raine, 36, daughter of the poet Craig Raine, has worked as a director and a playwright. After leaving Oxford, she got into directing and tentatively joined the young writers programme, feeling like "an infiltrator". She has a dreamy, self-deprecating intelligence. She characterises Cooke's rehearsal room as "one of the warmest I have been in. People wanted to do well for him." When she has been directing, he has always been "there with the first aid". Writing her own accomplished play – Tribes – he was "completely hands off" and would "just ring me up every now and then to ask, 'Any progress?'" She gives an apologetic laugh and cautiously admits she is working on a new play now, as well as directing William Boyd's adaptation of two Chekhov short stories.



Anya Reiss

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

Anya Reiss had her first play, Spur of the Moment, on stage at the Royal Court when she was 18. Susannah Clapp praised an "extreme young talent" and said she "puts on stage the jackdaw chatter, the fluffy vulnerability and the vulture behaviour of pre-teen chicks". Reiss, now a poised and unassuming 21-year-old, swiftly showed she was not going to stay a fledgling for long; her second play, The Acid Test, which she dismissively describes as "a post-university drunk-in-a-flat play", was widely praised too. She was so unused to the idea of herself at a playwright – at 18 – that she says she remembers thinking, at the first preview: "Oh shit … people I don't know are going to see this."



Bola Agbaje

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

Bola Agbaje, 31, thought the young writers programme would be like going back to school – she would get her work marked and then go back to her job as a housing officer. Instead, her first play, Gone Too Far! – set in Peckham, about sibling rivalry – picked up an Olivier. (She thought: "OK, this is getting serious.") Her second piece, Off the Endz, about criminalised black youth, stirred up debate. Agbaje's parents moved from Nigeria to the UK in the early 1980s and her third play, Belong, explored questions of identity and the African diaspora.

She is a mix of modesty and exuberance. "Dominic – I don't mean it in a cheesy way, but he is like an uncle to me…" What she has valued is his faith in her: "In this industry, he is one of the few directors who will say, 'OK, this isn't a finished product but I can see the diamond in the rough.'"



Penelope Skinner

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

At 30, Penelope Skinner thought she was too old for the young writers programme. But her debut, Fucked, at London's Old Red Lion got noticed and "they brought me into the fold". Now 35, she is witty and as sleek as a ballet dancer. She remembers when she and the other student playwrights locked themselves into the literary department and wrote through the night. "I suppose it will never happen again – it was a once-in-a-lifetime thing, like a weird festival. Leo, the tutor, was asleep on the sofa by two."

It was there that she wrote the first draft of The Village Bike. It was a slow process getting it on – it was turned down at first. But when it was staged, it got enthusiastic reviews. It was about a reversal of male-female stereotypes: the woman randily on the rampage, the boy a more homely proposition. She thinks the key thing to remember as a playwright is "write what you want and write from yourself".



Nick Payne

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

Nick Payne, 29, says before he joined the young writers programme he was "scrabbling in the dark" (and working in the National Theatre bookshop). He has the look of a helpful librarian and talks about how inspired he was by hearing a lecture at the Royal Court given by Ruth Little, then literary manager, arguing that plays should be treated as living organisms. Payne's first play had the wonderful title If There Is, I Haven't Found It Yet and won a George Devine award in 2009. Constellations used advanced physics as a metaphor for life and was intensely theatrical, a "move away from a kind of literalism in which you are asked to imagine almost everything". It transferred from the Royal Court to the West End. A film adaptation is in the works.

EV Crowe

Shot at the Royal Court: L - R: Penelope Skinner, Nick Payne, Nina Raine, Polly Stenham, Bola Agbaje, Emma Crowe, Dominic Cooke, Anya Reiss. Photograph: Suki Dhanda for the Observer

"I never thought the Royal Court would put on a play with trash-talking 10-year-olds to be honest …" Emma Crowe, 32, is talking about Kin, her disturbing, eloquent play about feral little maids from boarding school. She is impressively eloquent in person too. "I felt it was a big personal risk and thought the theatre was probably taking a risk too." This was followed by Hero – about straight and gay teachers. She has relished being produced at the Royal Court because of a "belief and camaraderie" that seems even to extend to "the people banging a nail into the set". She has been "quietly encouraged to cultivate my own obsessions and treat them with respect".

She has an attractive earnestness as she says that the Royal Court as a theatre reminds you that "there is some honour in an industry that is sometimes called showbiz". She studied at UEA and has done youth work, including teaching playwriting to 12-year-olds (some of whom are bound to be at the Royal Court themselves before long).

Mike Bartlett

From left: Laura Wade, Mike Bartlett, Bruce Norris (in front), DC Moore (Dave). Photograph: Suke Dhanda for the Observer

Bartlett, 32, entered his first play, My Child, for the young writers festival in the year Cooke took over. Bartlett was in HMV when Cooke rang to say: "We want to do your play – properly." It was "probably the best phone call of my life". My Child went on as part of an energetic season of first-time plays. And Bartlett's third play, Cock, about a gay man who falls in love with a woman, was, Cooke says, one of those rare plays that was word-perfect on submission. Bartlett explains: "Cock took the assumed moral that you are allowed to be gay or straight and went one further to ask: why are we pigeonholing people?"

Bartlett points out that Cooke's "legacy" has been partly about "delivering the right plays to the right audiences". He adds: "I can't write a good play without a social, political or economic idea at its heart, which I think fundamentally matters." He is a powerhouse of a writer – with work including the mind-bendingly ambitious Earthquakes in London at the National. He says: "Any play should be on the cutting edge of where we are morally."

Laura Wade

From left: Laura Wade, Mike Bartlett, Bruce Norris (in front), DC Moore (Dave). Photograph: Suke Dhanda for the Observer

Laura Wade, 35, is an incisive, graceful and withheld presence – in almost comic contrast to the content of Posh. She says it was slightly irritating for her that Posh was seen to be brilliantly opportunistic commissioning (because of Cameron having been a member of the Bullingdon club). It was assumed she sat down in February and thought: "Oh well, there is going to be an election in May" and then wrote it very quickly. In fact, she "had been slaving over it for three years".

Cooke had been shrewd about the potential audience on his doorstep: "He was interested that a lot of young people about the same age as my characters would sit out on the front steps of the Royal Court in the evenings, using it as a meeting spot for their friends, but never venture through the door." He sent Wade and director Lyndsey Turner "on a mission to find out about their lives".

She believes audiences want to see plays "that really question who you are, the choices you make [in] the way you behave towards others, the mark you leave on the world". Cooke said one thing that she keeps stuck on the wall in her study. "It is the key idea that a play should answer two questions: 'What is a play?' And: 'Who are we as people?'"

Bruce Norris

From left: Laura Wade, Mike Bartlett, Bruce Norris (in front), DC Moore (Dave). Photograph: Suke Dhanda for the Observer

Bruce Norris, 52, has been Cooke's most successful American import. A copy of his The Pain and the Itch was on Cooke's desk when he took over the job. "It was the first time a play of mine had been done in the UK. I was thrilled, overwhelmed and terrified." Norris is a wry, hilariously disparaging man. His Clybourne Park was an entertaining, cringe-making satire about bourgeois bigotry and the way that when people's property is threatened they become aggressive. He insists that because The Low Road, which opens on 27 March and is Cooke's Royal Court swansong, is about history and economics, it "does not sound like a rollicking evening in the theatre". But he adds: "Hopefully, it will be entertaining."