Although Beijing is as big as you expect – a dauntingly gridlocked, teeming, dust-laden bigness – it is only the third-largest city in China, behind Guangzhou and Shanghai. Yet, as the capital most likely to take over from Washing-ton, DC, as the world’s centre of supreme power, it has a hell of a swagger. When you see what Beijing is doing for the 2008 Olympics, you wonder why London is bothering.

Just as well we started Heathrow’s Terminal Five in good time, because, when Richard Rogers’s building opens next March, it will have taken nearly 20 years from initial concept to reality. In contrast, China decided it needed a vast new air terminal in 2003 and asked Norman Foster to design it. Then it brought in an army of 50,000 workers to build it, along with an extra runway. And it is finished, bar the final testing, which means it will open at the same time as Terminal Five. In front of the new air terminal is a large glass bubble, which turns out to be the railway station that will connect the airport to the city centre. “Do you know,” says Foster casually when I bump into him later, “you could fit Terminal Five inside that station alone?” As one-upmanship goes, that takes a whole barrel of biscuits.

Foster was in town, along with the sculptor Anish Kapoor, the former French prime minister Dominique de Villepin and a million art dealers and collectors and curators and auctioneers, for the opening of the Ullens Centre for Contemporary Art (UCCA). This is a rather fine converted factory at the centre of Beijing’s thriving art-production district. The first big modern-art gallery in China, it is a response to the way the country’s art has become very, very funky over the past 20 years. Wealthy entrepreneurs Guy and Myriam Ullens have been buying this art for a lot of that time, and have now gone a lot further by hiring the French architect Jean-Michel Wilmotte to turn a Mao-era armaments factory, built by comradely East German engineers at the top of their game, into something potentially very good. The opening show is a bit pedestrian, though. There are rumours of big fallings-out among the curators in consequence.

Prices for good Chinese contemporary art are now at western levels, which is to say absurd, if not quite Damien Hirst. Artists here drive around in swanky cars and cut fat-cat deals, often directly. Charles Saatchi has bought into it, and western artists such as Kapoor are rushing to exhibit in the edge-of-town art district – he has a show on in one of the bigger commercial galleries there – because it’s an important market. It’s like anything else made in China: efficiently produced, often clever, wonderfully exportable. The world’s museums are taking note. Next year, in Britain, exhibitions on Chinese art, architecture and design will blossom everywhere, from the Serpentine Gallery to the V&A. So you get that sense of being at a new centre culturally, as well as commercially and politically.

For their public buildings and spaces, the Chinese always take the grandly symmetrical approach. The Olympic park, in the northern sector of the city, is arranged on just such a rigidly disciplinarian axis, with various big lumps of sports buildings lined up like army cadets on parade. Not that many people are going to mind, because they will all be gazing at something so wildly improbable, it steals the show completely: the main Olympic stadium by the Swiss architects Herzog & de Meuron. Nobody could claim this crazy steel bird’s nest is remotely functionalist. It does the job of containing a stadium, but stadiums as a rule are dull. Designed in concert with the British engineers Arup, it is a seemingly random but, in fact, precisely calculated basket of steel noodles. As with the Foster airport, the closer you get, the more extraordinary it becomes. Can this thing possibly exist, right in front of your nose? In China, it can. The stadium is the world’s biggest scribble, drawn in free space.

Artists in the city are already adopting it somewhat literally, in one gallery showing it perched in a giant tree. No doubt, in the edge-of-city avant-garde community, someone else will have thought to show it with a giant egg laid inside it. In which case, the egg in question will be the city’s new National Grand Theatre, by the French architect Paul Andreu, best known for airports, notably Paris Charles de Gaulle. The theatre is an object so relentlessly, geometrically pure, it almost hurts. One block west of Tiananmen Square, it is not so much an egg as a giant soap bubble, but made of titanium and glass, and costing £180m. In the West, it would cost at least three times as much. Inside are three auditoriums – opera house, concert hall, theatre – that together seat 5,500 people. Surrounded by a square moat, disappointingly drained dry when I dropped by, it clearly sets itself up in comparison with the nearby moated Forbidden City. This is a dangerous game to play, and Andreu’s slippery dome loses the contest. There is nothing that says what it is or what it does. It could be the Communist party HQ, a nuclear reactor, anything. It is a wonder, all right – a thin scattering of people line up to be photographed in front of it – but as photo opportunities go, it doesn’t compare with that portrait of Mao looking out over Tiananmen Square.

Arup’s Cecil Balmond, master of the informal, complex structure, pops up again helping the Dutch superstar Rem Koolhaas build the knotted skyscraper of the CCTV building in the northeast of the city. It might sound like a security camera, but this is China’s state broadcaster. Koolhaas does not do polite architecture; he rejoices in the perversely difficult. This makes CCTV interesting, though I suspect it is more dramatic right now, half built, than it will be when finished. Its two angled legs are rising into the air, getting ever closer. They are starting to grope for each other like blindfolded party guests. There they stand, implausibly poised in the Beijing smog. In the end, though, it’s just another object-building by another superstar European architect.

Everyone wants to build in China, and these state-sponsored projects are as plum as such jobs come. At least the Ullens gallery is, at about £5m, a relatively low-budget conversion of an existing building in the middle of the artistic community it serves and feeds. Had this been left to the state, you imagine it would have cleared a slum and plonked down a monumental container amid hectares of empty space. As it is, the arrival of UCCA effectively saved the hugger-mugger arts district of former factories from being cleared to house yet more rampant commercial sprawl. Let’s be thankful to the Ullenses for suggesting, ever so subtly, that there is another, more incremental way to develop a world city.

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