It is like some kind of aesthetic in-joke: every two years, the world's most anachronistic city, dripping in the decadent trimmings and old world opulence of its golden age, also plays host to the most important contemporary art festival of our times. In the space of a few metres, you can move from a sumptuous palazzo on the Grand Canal to an installation of a car hanging from a church ceiling. You try to flag down a gondolier; he happens to be wearing black and has a coffin on board. A performance artist from Macedonia, of course.

The Venice Biennale is replete with such visual contradictions. As if the maze of alleys and waterways were not enough to disorientate the unsuspecting visitor, the attempt to present a definitive picture of contemporary trends in art completes the job. Last time around, in summer 2003, nearly 300 artists were displayed in locations around the city. Video screens flashed, performance artists wailed, installations puzzled. Anyone trying to make sense of it all would be better off trying to work out the city's notoriously opaque house numbering system.

Enter the Biennale's new president, Davide Croff, a banker who has held senior management positions at Fiat and the Banca Nazionale del Lavoro, who concedes that the festival has perhaps become "not very easy to understand". Bringing a touch of rigorous business thinking to proceedings, he has announced a programme for the next three years, starting with next month's edition of the festival, which he hopes will present "a clearer picture of where we are, and how we got there" to the world.

This year's 51st Biennale, directed by two Spanish curators, María de Corral and Rosa Martínez, (the first co-curatorship, and the first women to do the job), is already much slimmed down in terms of artists involved: just 91 contenders will bring their work to the city. Following the conclusion of the festival, a series of seminars in other Biennale centres - Istanbul, São Paulo, Shanghai - will lead to a grand symposium to be held in Venice in December.

The following Biennale, of 2007, will take account of the symposium's findings and present, Croff hopes, a state-of-the-art snapshot in a more coherent way than has previously been the case. "Contemporary art has become more complicated," he says from his palazzo office in the Campo San Polo. "With the growing number of languages used by artists, the effects of globalisation, and the major world events of recent history, things are more confused. We do not have the presumption of wanting to explain everything, but just to say, OK, let's try to reorder."

I put to him that the confusion of contemporary art trends is surely endemic to the modern world, and he is quick to agree. "Yes, but even if it is endemic, and we have to live with confusion, even just to think about it is something that would give added value to the discipline. Our ambition is to create a debate. And to be criticised - that is part of the game."

So by the end of the 2007 Biennale, some kind of value judgment will be expressed on the state of contemporary art? "Exactly. But this is not an original idea. The Biennale has historically been concerned not only to display the major tendencies in art of the time, but also to anticipate. That has always been part of its cultural mission."

He says this year's two curators will complement each other's choices: de Corral will present a show on The Experience of Art in the Giardini, the historical centre of the Biennale, which will survey present-day trends. At the Arsenale, site of the city's 14th-century armaments factory, Martínez will curate Always a Little Further, which will be more forward-looking. "It is partly to do with their personalities," says Croff of the two Spaniards. "María is more mature, perhaps more institutional, she has been at the Reina Sofia and so on, while Rosa is more aggressive and independent. They will both look forward, but from two different points of view."

Croff's banking background is no accident. The Biennale's other central concern is to widen its revenue base, specifically to bring in more private money. A reform last year that turned the Biennale into a foundation will give it greater autonomy to this end, says Croff. "But we are not particularly facilitated by Italian legislation. Private individuals receive little or no fiscal benefit [from donating to the arts], and corporations get marginal relief, which is totally different from a country like the US."

He says he wants to raise the amount of private money in the Biennale's €30m (£21m) budget, from its current 35 per cent of the total to 45 per cent, not least to protect it from a declining public subsidy. The Biennale is also organisationally responsible for the city's other arts festivals, celebrating music, theatre, architecture, dance and, most famously, film, and is showing some signs of strain.

But it is imperative, says Venetian-born Croff, to preserve the link between the city and its festivals of various art forms: "Biennale is a great asset to Venice, to Italy, to the world. Venice has a glorious past, but at the same time it is able to be at the frontiers of innovation. It is an opportunity not to be just a monument, but to live in the present, and look towards the future."

The 31 pavilions in the Giardini form a snapshot of the world order as it stood in the first years of the last century, with some of the buildings lauded as architectural marvels in their own right. But that world order is much changed today, and this year sees the inauguration of the Chinese pavilion, which will be located in the Arsenale ("The Giardini are full," explains Croff matter-of-factly.)

He pays the now-obligatory tribute to China's emergence as a superpower, and hints that culture will become a powerful platform for the country to express itself. "This year we were approached in a very strong way by the Chinese. They said that they were ready to participate, and to have a strong presence in the Biennale. A delegation of 20 people came to stay in Venice for a week. We were happy to talk about their participation, maybe for 2007, but they were very determined to come in now."

Croff also talks of exporting the Biennale in some way to China, and indeed to other world venues. He says a tour of the 2003 exhibition to seven cities in southern Italy was very successful in attracting new audiences. Was he trying to create a brand? "In a way. Globalisation is total in this business. There are people who cannot come to Venice, so we can take Venice to them."

I ask him finally about the city's infrastructure - could it cope with the ever-expanding visitor numbers to the various festivals? There was an outcry at last year's film festival, when the film schedule overran and Al Pacino was forced to wait for a couple of hours for the delayed premiere of The Merchant of Venice.

Croff says he is looking for a big new venue in the city that could act as a headquarters and cultural centre, and then sighs and shows just a touch of exasperation. "Sometimes I am very puzzled by the press. We talk about big projects and great ideas, and all they focus on is the lack of mineral water, or the fact that it is too hot. Yes, it's true about Al Pacino. He was in my office. But he was happy." Venice, he might have added, has that effect.