It sounded like a cross between a joke and a nightmare but nothing, surely, to do with art. On Tuesday evening, 150 schoolchildren from all over the country travelled to Tate Modern to spend the night in the gallery, in a giant “sleepover”. They slept in tents, strewn over the floor of the gallery’s vast Turbine Hall, which they had painted themselves, inspired by some of the works they had seen during the day.

One imagines that a lot of fun was had. But it doesn’t end there. The sleepover was the beginning of a process that will culminate in a conference in the autumn of 2008, entitled “From My Space to Your Space.” It will be run entirely by young people, who will decide whether to invite adults or not, and if they do, who they might be. At the end of the conference, they will produce a “Creative Manifesto”, which will contain their views on what they like or don’t like about the gallery, and what they would like to see there.

An innocuous piece of educational outreach? Anything but. It so happens that Tate Modern is building a dramatic new extension by Herzog & de Meuron, at the cost of a cool £215m, next to the mother gallery on the south bank of the Thames. And it seems that what the young people want, within reason, they are going to get.

Tate’s director, Sir Nicholas Serota, admitted as much at a press conference unveiling the gallery’s long-term plans last month. “We are building a gallery for the next generation – and we want to listen to find out what it wants to see in it,” he said. Tate Modern was called that for a reason, he added; it was not simply an art gallery or a museum, but something more. Something truly modern.

The chances are that, when the new building opens some time before the London Olympic Games in 2012, it will bear little resemblance to the traditional art gallery as we know it. The existing Tate Modern has already started that process, replacing the orthodox art historical, chronological hang of paintings with an arrangement that requires lateral leaps of the imagination: you view the abstract, rectilinear grids of a Mondrian masterpiece through the sensual curves of a Brancusi sculpture; you step on a Carl Andre floorpiece to get a better view of a Kandinsky painting.

But the Tate of the future will go much further. Already, it is venturing into new territory that is explicitly aimed at attracting the 16-24-year-old generation. Tate Tracks commissions bands such as The Chemical Brothers and Klaxons to write songs based on works in the gallery, and then plays them exclusively on listening posts next to the works.

If you are one of the 1.4m monthly unique visitors to Tate’s website, you are used to Tate Shots, short, downloadable videos based on some of the gallery’s artists and events. In perhaps the gallery’s weirdest collaboration, it is teaming up with football’s Premiership for its Kickz campaign, using art as well as football to engage with children from disadvantaged backgrounds.

In its spaces, and especially in the Turbine Hall, Tate Modern has wilfully blurred the boundaries between art and plain fun. Carsten Höller’s slides in the Turbine Hall helped attract a record numbers of visitors over the past year (the gallery now receives about 5m spectators a year, making it the most popular modern art gallery in the world). At last week’s UBS Long Weekend, a carousel surrounded by tables laden with toffee apples was described as an art work by the Brazilian artist Marepe, using “found objects” from his native country. But it was, indubitably, a carousel surrounded by toffee apples. Both pieces would have looked more than comfortable in a funfair.

So this is Tate Modern today: groovy website, playground, social worker, performance space. What, we may ask, of the art? What of the intimate moment of connection with an individual work that has the power to change lives? Well, it’s there, in theory, if you want it. There are some great paintings in Tate Modern. But it is disingenuous to pretend that they are not being trampled underfoot in the clamour all around them. Tate Modern was not called the Tate Museum of Modern Art (which is just as well, as critics rushed to point out the deficiencies of its collection). It was always meant to be something more.

Or, perhaps, less. Traditionalists decry what has happened at Tate Modern as some kind of weird aberration, a vanity project by the all-powerful Serota, the ultimate statement of the dumbing down of British culture. But they are missing something. What is happening at Tate is happening across the boundaries of the British cultural scene. And it is what is making London the undisputed cultural capital of the 21st century.

When I interviewed Antony Gormley in these pages last month about his exhibition at the Hayward Gallery, he talked of a mischievous desire to place CCTV cameras on Waterloo Bridge, in an attempt to make the show “bleed” into the streets around the gallery. It is this bleeding of culture, into streets, squares, sitting rooms and ordinary people’s everyday lives, that is its defining characteristic at the beginning of the new millennium.

It is almost as if the traditional temples of high culture, the galleries, museums, concert halls and opera houses, could no longer bear the load of having to improve people’s lives, providing them with spiritual nourishment as the world around them became faster and shriller. The pressure was too intense; they simply exploded out into that world and, in some cases, became as fast and shrill as the universe from which they were supposed to provide refuge.

Everywhere you look, in some form or other, there is this melding of cultural and everyday life. On Tuesday, the Royal Festival Hall unveiled its magnificent £111m facelift to the world; in a telling aside, the Southbank Centre’s chief executive Michael Lynch earnestly explained how he had been in constant negotiations with the skateboarders who used the underground spaces down below, out of respect for their activities. (The Royal Festival Hall was always meant to be democratic; but that democratic?)

On Wednesday, the Royal Opera House announced its ground-breaking purchase of the Opus Arte DVD label, with the mission, sitting comfortably alongside its intention to make money out of it, of spreading its productions around the globe, via the variety of new digital media that suddenly make such aspirations possible: the highest of art forms transported into the humblest of sitting rooms.

Let’s go to the theatre; or rather, let’s not. Next week, the performance duo Lone Twin are playing at the Barbican Centre as part of the international BITE theatre festival. Or at least, they will get to the Barbican eventually. The performance consists of the two men drawing a spiral from the nearby Underground station to the centre, and journeying along that line, regardless (up to a point) of physical impediment. Once more, an art form breaks out of its presumed home.

There is culture in my pocket: my Oyster card holder is designed by Tracey Emin. There is culture in my shopping trolley: my Sainsbury’s plastic bag is by Anya Gallaccio. (Both courtesy of the Arts Council’s 60th anniversary.) Culture seems to seep, or bleed, into every corner of contemporary life.

The notion that art is for everyone is not a new one. It was one of the founding principles of the Royal Festival Hall, the Edinburgh festival, the BBC’s Third Programme, all laudable, idealistic, post-war projects rooted in a desire to use culture as a means to prevent future conflict. But it was a highly specific type of culture. It was sophisticated, complicated, demanding. It was never going to appeal to everyone; especially when everyone discovered television and The Beatles. High culture was ambushed by pop culture and mass entertainment.

But gradually, thanks to the demographic shift that brought the baby boomers to power, that distinction has become meaningless. The issue of accessibility scarcely applies any longer: all our great cultural institutions, guided by some very shrewd and enlightened leaders, have gone to extraordinary efforts to make their work available to all. (The cheapest seats at Covent Garden are considerably cheaper than their equivalent at Stamford Bridge, or at a Barbra Streisand concert.)

Art forms, high or low, have become promiscuous. There are musicals at the opera, operas in shopping malls, shops in art galleries, art displays in department stores. They have all become part of a virtuous circle that enables money, art and people’s spare time to feed off each other.

There are of course those who think that all the above marks something like the collapse of civilisation; that this feckless, playful, demagogic entertainment can and should never take the place of “proper” art. Yet here is the beauty: it does not need to. If you believe in the traditional relationship between the arts and their consumers, that demands concentration, intimacy, study and time, it is all still there, as resplendent as ever.

There is the British Museum, the National Gallery, Tate Britain, the Wigmore Hall, the Royal Shakespeare Company, the BBC Proms and so on. Go to the Wallace Collection or the Courtauld Institute of a weekday afternoon, and you will have your safe haven of sacred quietude. The Royal Opera puts on Massenet’s relatively obscure Thaïs in Concert this summer, and it sells out in a flash. Blockbuster shows of Old Master paintings can be the hottest ticket in town. The relationship between traditional culture and the culture of the new millennium is not a zero-sum game. They spur each other on. Everyone’s a winner.

We have never been surrounded by so much culture. Some of us may think that some of it doesn’t amount to much; but that debate alone is worth the initial investment. It makes for the kind of robust exchange of views that is the lifeblood of any democracy. Where will it all take us? It is too early to say. That is up to the next generation, represented by the Tate Sleepover gang, who will have been brought up to believe that culture is malleable, exciting, free, and all around us. And that’s not a bad thing to think, is it?

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