" The conversation we're having is about women leaders in the arts: the relatively recent influx of women to artistic director positions in theatre, and the regularity with which women are directing main-stage plays for the National and RSC, persuades Heywood that women can finally claim equality with men.

She's not alone in her thinking. In her 20 years working at Tate, Frances Morris, now head of international collections, says she has witnessed "a steady takeover bid by women", particularly in curatorial departments. In fact, says design critic Alice Rawsthorn, the visual arts industry is now "predominantly female". At the Royal Opera House, Elaine Padmore, head of the Royal Opera, and Monica Mason, director of the Royal Ballet, argue that management positions are split between women and men. Since she became director of the British Film Institute in 2003, Amanda Nevill has been "surrounding myself with brilliant women. As an executive, we say we need more men to balance us."

And yet, in April, culture minister Margaret Hodge slammed the arts industries for a lack of diversity, particularly of women, in executive and non-executive positions. Shortly afterwards, the Sunday Telegraph published a list of the 100 most powerful people in British culture: just 18 were women, and only three appeared in the top 20 (Vivien Duffield, JK Rowling and Julia Peyton-Jones). Talking to women working in the arts - I talked to 23 for the purposes of writing this article - several stories of inequality emerge. "I can't tell you the number of times that I go into meetings where I'm the only woman," says Nevill. At last year's Society of London Theatre Christmas lunch, theatre producer Sonia Friedman estimates that she was one of five women among 80 men. "I remember sitting down at a table full of men saying, 'This is pathetic representation.'"

Nadia Stern, chief executive of Rambert Dance Company, says, "Margaret Hodge is right, and it's a bit depressing that this is still newsworthy. Dance is still very male-dominated: most of the choreographers are male; most of the designers are male; when I meet my counterparts in venues around the country, they are almost always male. Given the talent that is out there, something is going on if most of the people in those positions are men."

So what is the real situation for women in the arts? There are almost as many answers as there are women. Personal experience counts for much: women working in the "feminised" visual arts have a different attitude from those in, say, the music industry, which Jeanette Lee, co-owner of Rough Trade Records, says is "run in quite a male way, like a team sport". As Virginia Tandy, director of Manchester City Galleries and co-founder of the Women Leaders in Museums Network, says: "If you're the female director of an independent arts organisation with a substantial number of women on your management team and, hopefully, a substantial number of women on your board, what's the problem? But if, as I have, you've sat on the board of a national organisation and there have been three women and 14 men, you think: this isn't right."

Boards are definitely an issue: a paper published in March by Demos and London's City University found that women are not only under-represented among trustees, but may account for as few as 28% of board chairs. There is some correlation between this fact and another: that, while it's possible to reel off lists of names of women who work in the Arts Council England, as the heads of medium-scale institutions or arts companies, or in middle-management, once you look at what theatre producer Emma Stenning calls the "headline jobs", they are chiefly occupied by men.

Women are breaking into these upper echelons, notably Jude Kelly as artistic director of the Southbank Centre, and Diane Lees, who takes over as director-general of the Imperial War Museum in October. But Stern sounds a cautionary note: "You can't judge the norm by the exceptional people. It would be like saying America's not racist because there's Condoleezza Rice."

Talk to enough successful women, and you start thinking that women are on an equal footing with men. And yet it's difficult to reconcile that with the fact that fewer than 25% of British theatres have female artistic directors, that Kathryn McDowell is the only managing director of a British orchestra (the London Symphony), or that four of the 14 senior staff at Tate are women. No woman would want to be given a job simply to push these numbers up. "What we want is not x percentage of women in senior jobs," says Lees. "What we want is a sustainable sector that has the best talents leading it at whatever level." Theatre director Thea Sharrock puts it more bluntly: "If a women isn't the right person for the job, don't fucking give it to her. I'd much rather have a rash of good men running buildings than the wrong women."

There is a widespread belief that the "who you know" networking of old is at least being balanced, if not entirely replaced, by meritocracy, and that reaching the top is now, in McDowell's evaluation, "a matter of choice". Lees points out that there is "a difference between a forced block and a choice block", and what women primarily face is the latter. If that is the case, why might women choose to stop in the middle of the ladder?

There are a few key reasons, not all of them exclusive either to women or to the arts. One is the conflict between success and creativity. Vicky Featherstone, artistic director and chief executive of the National Theatre of Scotland, says she has had to take a year off from directing shows herself to concentrate on running the company. In a sense, theatre is a special case: an artistic director, Heywood says, is "supposed to be Picasso and build your own art gallery".

By contrast, Morris says she has benefited from the expansion of the arts industry, which has meant that heads of collections are no longer expected to manage budgets. But she also recognises that "there is a danger in more senior positions at Tate that the creative involvement suffers". Lees concurs: "I've done one exhibition in eight years. That's the interesting thing about leadership: you have to make a decision about which direction you're going to take. There's a lot to be said for the people who decide not to become the big museum director, but to lead their field from a subject-specialist point of view."

Then there's the fact that success consumes entire lives. "Big jobs are demanding jobs," says Andrea Nixon, executive director of Tate Liverpool. "There may be a point where people say, 'There are only so many hours in the day, including my weekends, I wish to commit to work.'" No one would claim that long hours are unique to the arts: running an institution may be "unbelievably self-sacrificing", says Susanna Eastburn, director of music strategy for Arts Council England, but we shouldn't forget that "presenteeism culture is endemic".

However, the industry can make peculiar claims on people. Alison Donald, managing director of A&R at Chrysalis Music Publishing, suggests few women maintain a career in the music industry because "the hours are not conducive to a normal social life: I'm in the office all day, and then I'm either out seeing bands, or having dinner with a manager". And, unlike other professions, those who work in the arts rarely find that their long hours are recognised in their payslips. "Very, very, very few people make a comfortable living in the arts," says Stern.

Long hours and low pay are particularly problematic for women who have children. It's not impossible to be a mother and hold down a high-powered job: Featherstone has two children, aged eight and six, and says the balance is "mind-breakingly challenging".

But many women don't even attempt it: of the 23 I talk to, 11 don't have children. "I do not know how I would do what I do and have a family to look after," says Friedman. "Producing is a 24-hour job. It's not even a job, it's a way of life. And there are probably far more men doing it than women, because men aren't the ones at home bringing up the children."

Asked why the long hours don't seem to affect men in the music industry, Donald snorts, "Because they've got wives!" Featherstone bristles when asked about her work-life balance because, she says, "I never hear men asked that question. There is an assumption that there will be a structure behind men, and that structure is women." But Sharrock, who recently gave birth to her second child, feels that one can't compare men's and women's experiences when it comes to children, for "a very obvious primeval reason: it's just not their body. It mustn't be a handicap for women, but it has to be acknowledged." Lees, who doesn't have children, agrees. "Until men have babies and that balance shifts, women are going to be self-limiting by the choice to have children."

Polly Teale, who co-runs the theatre company Shared Experience, says that when her husband Ian Rickson was artistic director of the Royal Court, the pair felt that "we both needed a wife, in the old-fashioned sense". Joking aside, she wonders whether leadership positions need to be restructured to allow women to maintain those jobs alongside a family life. Most women find themselves working in what the choreographer Siobhan Davies describes evocatively as "man-made architectures". If more women reached positions of power, could that architecture be rebuilt? The notion makes Stern laugh: "Look what happened under Thatcher!"

To a degree, long hours, low pay and the work-life imbalance are not gender-specific problems. Depressingly, the problem that seems to be unique to women is their own self-doubt. This sounds perilously like a stereotype-ridden generalisation, but Rawsthorn, who regularly sits on arts appointments panels, offers worrying evidence of its truth. All candidates, she says, are psychometrically tested, and "you can always guess whether a candidate is male or female. The characteristics that are flagged up again and again for the women are: self-doubt, questioning their ability to lead, and reluctance to step forward for promotion. It all adds up to a weaker sense of female entitlement for those very senior jobs."

Lees, another co-founder of the Women Leaders in Museums Network, says it was partly to counter this sort of thinking that the network was set up. She is interested in the idea of self-limitation because "a lot of my experience has been about ruling myself out of an opportunity. There have been a couple of jobs in the past five years that I would now say I was mad not to go for."

It's fascinating how many women hold themselves back in this way. When Eastburn, a former artistic director of the Huddersfield contemporary music festival, saw that job advertised, she assumed she couldn't do it, and applied only because friends persuaded her otherwise. Kathleen Soriano, now director of Compton Verney, spent 17 years working as the exhibitions manager at the National Portrait Gallery, convinced she could never run an institution herself. It wasn't until she became a fellow of the Clore Leadership Programme, set up in 2004 to expand the pool of potential arts leaders, that she stopped seeing such leaders as "godlike" and discovered the confidence to take a step up.

Of course, ambition and confidence are as much about personality as gender. It's instructive to look at the different paths taken by Morris and Iwona Blazwick, who were appointed together as the first programme curators at Tate Modern. While Morris has quietly continued her career at Tate, Blazwick is the high-profile director of the Whitechapel Gallery in London. "Iwona is a leader," says Morris, "and I would say I'm a team leader. And maybe I'm a team leader because I've spent the past 17 years helping to lead a family team." Morris also feels that the education of girls is no longer as gendered as it was 20 years ago. "I look at my daughter, who's 17, who's studying art but also economics, and I think: you could run the world!" Nevill has a similar feeling about her own daughters, now in their early 20s. "They don't create barriers for themselves, because they've never seen barriers there in the first place."

It took several decades of feminist discourse for women to infiltrate the workplace. Men still have the weight of history on their side: ask Padmore why there are so few women composers and she points out that "it is still almost within our lifetimes that it is normal for composers to be female". We're already at the stage where Featherstone, who turned 41 last month, can say that opportunities for women "have been a long time coming in the history of the world, but in the history of my life it's been open doors all the way". The future, thinks Lees, could even see male leaders in a minority. "Because we've got this massive middle-management tier of women in museums now, the pool that's available for the big jobs could be only women."

Some women think there are now other, more pressing diversity issues to be addressed. Heywood highlights the frustration felt by black and ethnic minority theatre directors, whose opportunities to work in the mainstream are few. Rawsthorn suggests that low pay is enforcing a class divide in the arts. Kate Horton, executive director of the Royal Court, agrees: "Class is the big issue.

I'm prouder of being a Brummie running the Royal Court than I am of being a woman."

When such inequalities are rife, does it matter if women reach the very top - the "headline jobs"? Of course it does. Some of the reasons are quite thorny, threatening to drift into essentialist generalisation again: there is the possibility that women will champion the work of other women, and are more likely to engage with challenging feminist work. There is the possibility of a difference in women's management style, which could be said to be more empathetic and team-oriented than men's. It's also important to enforce the idea that there are no barriers for younger women by providing role models.

Perhaps, though, it matters simply because women should have full equality for the sake of it. Emma Dexter, director of exhibitions at Timothy Taylor Gallery, puts it best: "It matters in the same way that it matters whether there is a black or woman president in the United States. Once we've had parity for 20 or 30 years, then you can say it's irrelevant. But that hasn't happened yet".

Promote that woman!

Theatre

Thea Sharrock, has already run Southwark Playhouse and the Gate in London, and she has shown a sure commercial touch with the West End revival of Equus (starring a naked Daniel Radcliff e). Given her poise, assurance and talent, it is only a matter of time before she takes over one of the theatrical citadels. Earlier this year, Josie Rourke ran a brilliant political campaign when the Bush in London, where she is artistic director, was threatened with a damaging grant cut. And, even if her initial play choices have proved erratic, she looks like a multi-skilled all-rounder. She learned her craft under Michael Grandage at the Donmar and has something of her mentor's versatility. Under the charm there is also a hint of steel, as her battle for the Bush proved, which should take her to the top.
Michael Billington

Visual art

Tipping Iwona Blazwick OBE, as the next director of Tate is what you might call a no-brainer. Currently running the Whitechapel, a job that Nicholas Serota held before taking over Tate, she has made a great success of it, with striking exhibitions of artists ranging from Robert Crumb to Paul McCarthy. If Blazwick should miss the top job, one of her rivals would be the excellent Frances Morris, whose credits include the collection displays at Tate Modern and many exhibitions there, not least its recent Louise Bourgeois retrospective.
Jonathan Jones

Dance

At 33, Tamara Rojo, is still far from ready to give up the stage, but she has made it clear she sees her future as an artistic director. The likelihood is that she will return home to Spain, but if the 12 years that she has spent in the UK encourage her to stay, this articulate, intellectually curious ballerina would be a stellar successor to Wayne Eagling at English National Ballet. If ever Mark Baldwin should want to move on as artistic director of Rambert Dance Company, Assis Carreiro would make an interesting replacement. She is an extraordinary dynamo and has turned Dance East from a small regional agency into a powerhouse of innovation and activity.
Judith Mackrell

Classical music

Mary Miller, has been away from these shores since her time with the English National Opera Studio (she is responsible for Stavanger's current stint as this year's European Capital of Culture alongside Liverpool), and it's about time we had her back. The Edinburgh International festival would make the most of her experience across all the art forms. One for the future: Susanna Eastburn, director of music strategy at Arts Council England, brings her experience as artistic director of the Huddersfi eld Contemporary Music festival and a high-powered job with the London International Festival of Theatre. After what will no doubt be a long stint from Roger Wright, she would be perfectly placed to take over as director of the BBC Proms.
Tom Service