We think of Bobby Fischer, the paranoid anti-Semite raging at imagined enemies, or Paul Morphy, the 19th-century champion, found dead in his bath surrounded by women's shoes. We recall Wilhelm Steinitz, the Austrian who thought he was in electrical communication with God. Even 'normal' professionals, like our own Nigel Short and Jonathan Speelman, tend to betray curious eccentricities. Short speaks English as though it were a foreign language and Speelman looks, in his clothes and grooming, like a mad professor plotting to blow up the world.

It should be said in his defence that Kasparov, though a genius at the game, has never particularly looked the part. Whereas Anatoly Karpov, the man he beat to take the world title back in 1985, possessed the cadaverous pallor and puny physique of someone who had spent his life in airless rooms lifting nothing heavier than a wooden chess piece, Kasparov by contrast was all simian movements and saturnine stares. His opponents spoke of his physical presence as if he were an athlete or a boxer rather than what they were, chess nerds. He trained for big matches with a fitness regime and careful diet. Short referred to his 'weightlifter's energy' and that was before Kasparov crushed him in their 1993 World Championship match.

And while chess players tend by nature to be self-absorbed, distracted, reclusive, Kasparov was an extrovert with no interest in concealing his high opinion of himself. As he put it in his 1987 autobiography, Child of Change, he was 'not one to hide my light under a bushel'. In the book he described his triumphs with an undisguised pride bordering on glee. He quoted praise at length and gave short shrift to criticism. Dubbed by his rivals the 'Beast of Baku', he later outlined his relationship with his fellow chess professionals. 'Most of the other players hate me because I beat them regularly,' he explained. 'Most of them have a devastatingly bad record against me.'

Everything about Kasparov, including his impenetrably thick hair, seemed to speak of the indomitable. He produced moves that teams of grandmasters would take days to unravel. And for 20 years, give or take the occasional blip, he could not be beaten. Then, earlier this year, having won the prestigious Linares tournament in Spain, he announced his retirement. Not for him the obscurity or notoriety that his predecessors encountered after chess, not for him the struggle against dwindling powers and memory. Instead, Kasparov informed the world that he was ready for a whole new challenge. He was going into politics.

It was a decision that underwhelmed many Russian observers. A column in the Moscow Times outlined some of the doubts about Kasparov's political abilities. 'The country's media, political analysts and even some of his fellow liberals see him more as a dilettante who does not understand the rules of the game and who has more than one failed political venture to his name, from the Democratic Party of Russia, to the Liberal-Conservative Union, and now to the risky Committee 2008: Free Choice. Critics and even friends of Kasparov have noted an inability to commit to any one project for a sustained period. In short, everyone seems to be telling Kasparov: "Don't dabble with the real world, go back to the safe confines of the 64 squares on the chessboard." '

Kasparov once quoted his countryman, Vladimir Nabokov's appraisal of the talents required to excel at his chosen game. 'Chess problems demand from the composer the same virtues that characterise all worthwhile art: originality, invention, conciseness, harmony, complexity - and splendid insincerity.'

It's safe to say that in the art of politics you can do without all of those talents except, of course, the last. And not even his fiercest critics would accuse Kasparov of insincerity, splendid or otherwise. In an interview two years ago, he seemed to acknowledge that his need to voice unpalatable truths might be a weakness in the political arena. 'With strong views and very little flexibility, you don't make a good politician,' he said, before adding a tantalising rider. 'Except in some crucial moments.'

It just so happens that Kasparov believes that now is one such crucial moment. 'In Russia,' he told me, 'there is no more politics in terms of campaigning, promoting your views, debating your points. The regime has destroyed democratic institutions. In such circumstances what you need is to raise your voice, stand firm and fight the regime. I don't think flexibility helps you very much. It's about fighting.'

Kasparov was probably the most aggressive player in modern chess. Though nearly all chess players like to talk about 'destroying' or 'crushing' their opponent, very often the way they play can appear more like boring their opponent to a standstill. Not Kasparov. He relished a fight. He moved his pieces as if they were weapons. He once said that he learned about politics through chess, and if that's the case then Kasparov's current tactics make a kind of chess sense. He has decided to target the King. He wants to take out the president of Russia, Vladimir Putin. But to succeed he will have to play the game of his life against an opponent loaded with almost every advantage. When he was a small boy, he once recalled, he saw a chess game 'in which small pawns were victorious over what looked like a more powerful enemy. This captivated me, and I have loved to attack ever since childhood.' But for Kasparov to defeat the Putin regime would be more like the equivalent of a solitary white knight vanquishing a near full set of black pieces.

I met Kasparov back in June, in a small hotel discreetly tucked away between Knightsbridge and Chelsea. He was wearing a light-green checked suit and a cravat that made it appear as if his wardrobe selection was the fruit of judicious reference to the works of PG Wodehouse. He sat on the edge of his seat as he talked, just as he used to when playing chess.

There's a physical impatience about the man that suggests he's already three steps ahead of you and he may not care to wait. I went out to put some coins in the parking meter just before he was about to be photographed and when I returned a couple of minutes later the photo session had been completed.

Kasparov has always appeared in a rush, which may explain why he used to look so precociously middle aged. Born Garik Weinstein in Baku - what was then part of the Soviet Union but is now the independent state of Azerbaijan - Kasparov lost his father when he was seven. 'I had to rely on myself,' he told me, though he also acknowledged the vital role of his mother. 'From an early age she decided to dedicate her life [to me], which made a very strong bond which affected my life - for good or bad, that's another story.'

There is a wealth of Freudian literature on the meaning of the Queen in chess, and many chess players, including Fischer (whose mother was called Regina) have had intense and often difficult relationships with their mothers. Similarly, in keeping with the Oedipal theme (the aim of chess is to kill the King), they have also, again like Fischer, often had absent fathers.

It has been written that Kasparov's father, who was Jewish, perished in a car accident. But along with the widely disseminated idea that Kasparov speaks 15 languages (he speaks two) this story lacks only the quality of truth. 'He died from leukaemia,' Kasparov said. 'We never had any money to buy a car.'

At the age of 10 he was taken under the wing of Mikhail Botvinnik, the Soviet former world chess champion. At 12 he was identified as a future world champion. By the age of 22, when he became world champion, Kasparov displayed the self-possession of a man of double those years. Photographs from the time show him dressed in a jacket and tie, with the kind of facial hair that requires shaving on the hour, and they make him seem almost eligible for a midlife crisis.

Now, at 42, his chronological age has at last caught up with his physical self. He has been married twice and has two children, he has travelled the globe and lived under both communism and capitalism and whatever system it is that now operates in Russia. But whatever world weariness he's accumulated is more than offset by his natural energy if not, perhaps, what you would call ebullience.

On the desk to his side there sat two books. One published by the Federation Internationale des Echecs (Fide) and the other titled Speak Like Churchill. Chess and politics, his twin passions. His great political hero is Winston Churchill. 'He came up with his ideas, he fought for them and his ideas were right,' he says. Kasparov may have some of Churchill's resolve, but he certainly lacks his oratory skill, at least in English. He talks politics in a dry analytical style that may, one hopes, have lost something in translation.

It's only when he conveys a sense of moral outrage that his words achieve a heroic flourish. Halfway through our conversation, which lasted for a couple of hours, he said: 'Recently I've been contemplating my future steps and I realised my choice is very simple. Either I leave my country or stay, and if I stay I have to fight. To ignore what has been done by KGB rulers would be damaging my sense of honour. Leaving my country would also be damaging because, why should I leave? Why me, why not Putin?'

His dislike of Putin is genuine. A dismissive sneer contorts his face each time he mentions his name. 'I think simply that the man doesn't fit his position,' is his opinion of the Russian president. And when I ask what he thinks is Putin's estimation of him, his response is withering. 'Frankly speaking, I don't care. I've met too many KGB colonels in my life to pay attention to their opinions even if they turn out by accident to be presidents of my country.'

The fact remains, however, that Putin's various moves against democracy and free speech have had little effect on his international reputation and even less on his domestic support. He may have closed down or bought up the overwhelming majority of critical media. He may have overseen a ruthless, in many ways disastrous, and arguably genocidal campaign in Chechnya. He may do favourable business with the oligarchs who back his regime and hound those - like the recently imprisoned Mikhail Khodorkovsky - who protest against it. But none of that overly concerns the mass of Russians for whom a strong authoritarian leader appears to remain a preferable path to the uncertainties of democracy and the free market. A serious opposition movement has simply not taken shape, partly because of Putin's grip on the media and partly because the high price of oil has buoyed a fragile economy. And in any case Putin is bound by law to step down in 2008.

Nevertheless, Kasparov is focusing on Putin because he believes that if he can expose him as a self-serving autocrat and bring him down, he can also strike a blow against the venal bureaucrats and billionaires who form a crony state. 'More and more Russians are realising that Putin is creating unique conditions for oligarchs to plunder the country and take money outside, providing they pay their dues to the regime. [Ramon] Abramovich,' he continues, citing the oil magnate and owner of Chelsea football club, 'is for Russians a symbol a corruption.'

So what, as another Russian once asked, is to be done?

'Travel across the country, meet people, tell them what's right, what's wrong. People like me are not allowed on Russian TV. The only solution, the partial solution is to go round and talk to people. In many regions there is still an open window of opportunity because local TV stations are still in private hands. I had 14 appearances in Siberia in two days.'

Kasparov told me that he was shortly off to the North Caucasus, where the Beslan atrocity took place. If he did not know before, then he was to see on that trip just what he was up against. He was prevented by the authorities from attending various meetings. He had eggs and ketchup thrown at him. In three cities his plane was refused permission to land. Hotels were mysteriously booked up or closed and venues were suddenly cancelled. It was clearly a concerted effort to stop Kasparov from spreading his message.

I asked Maria Lipman, a political specialist at Moscow's Carnegie Centre, how Kasparov is seen in Russia. The fact that he was half-Armenian and half-Jewish would not play well with Russian nationalists, she said, 'but it's really that he's so radical in being a westerniser and so clearly wants Russia as part of the globalised world that is of limited appeal to most Russians.'

She was impressed by the way that Kasparov had taken to travelling across the country. 'He seems to be really enjoying it. I don't think other liberal politicians have the energy or cash. They got disillusioned early on.'

All the same, she did not think that Kasparov's standing as a chess hero did much for his image in Russia. 'Chess has lost its popularity from Soviet days,' she explained. 'It's not longer promoted at the highest levels. And also even when he was world champion Kasparov was always seen as too anti-state, too cosmopolitan.'

This view of Kasparov as somehow an outsider goes right back to the days of his greatest triumph. In 1984 he challenged Anatoly Karpov, the world champion who inherited the title when Bobby Fischer retreated into a religious cult and refused to play him. Karpov was the darling of the Soviet establishment. A dedicated communist (he has since become a hard-core nationalist), he was treated like the official Soviet candidate.

Kasparov, too, was a communist, or at least a member of the Communist Party. 'Without making that step,' he told me, 'I wouldn't have got the support that was absolutely crucial for me to survive in that environment.' He later became an advocate of Mikhail Gorbachev, then an opponent, when he became frustrated at the slow pace of reform. It was not until January 1990, however, that he handed in his membership of the Communist Party.

Back in 1984, Kasparov showed no such deliberation. He tore into Karpov, launching a series of fearless attacks. As a result he lost three of the first seven games. Then two of the next 20 games. He was 0-5 down. Karpov had only to win one more game to retain his title. But Kasparov made an amazing and gruelling comeback, drawing game after game, and then winning three.

After five months of play, Karpov was a nervous wreck and looked close to exhaustion. His friend, Florencio Campomanes, the president of Fide, controversially decided to end the match just when Kasparov, for the first time in the event, looked to be the favourite to win.

Typically, Kasparov does not undersell his achievement. 'If you look at the odds of surviving in such a situation against Karpov, I don't think there's anything comparable in the history of any sport.' When I asked him the daunting size of the political task he has now undertaken, he replied: 'After saving that match against Karpov I believe anything can be done.'

Despite his anger with Fide, Kasparov won the rematch and so began his 20 years of dominance. It's true that he lost his title in London in 2000 to Vladimir Kramnik, but by then the chess World Championship had begun to resemble boxing, with competing governing bodies. In any case, Kasparov now says the defeat did him a favour 'because I could reinvent myself playing new, highly sophisticated chess and that gave me five years on top'. When he retired he was still rated number one in the world.

Kasparov, who writes a column for the Wall Street Journal and is busy with a series of books, will survive without chess (he restricts himself nowadays to anonymous games on the internet), but it's much less clear how well chess will do without Kasparov. There are no household names. He was the one player with global appeal and the professional game is unlikely to emerge from his considerable shadow for some time.

Equally uncertain is how Kasparov will fare in politics. He was renowned for his bold openings on the chessboard and so far, with his Russian roadshow, he has not disappointed. But does he have a middle game, much less an endgame? His various attempts to form a liberal coalition have failed, his critics claim, because of his unwillingness to compromise and his need to control decisions. And many observers expect his latest venture, the United Civil Front, which aims to combine all anti-Putin forces from left to right, to go the same way.

There is also the matter of financing. Kasparov is a wealthy man but not so wealthy that he can maintain a political movement without proper backing. 'We've got very little money,' he conceded, 'but the good thing is that there are quite a few rich people who are now ready to finance us anonymously. Funding us is a death warrant for any Russian businessman.'

So far, he has confined his stated ambitions to a negative - getting rid of the Putin regime. What he has refused to do is confirm his own interest in becoming president. I asked him about his long-term aim.

'Well, it might be that I find myself very useful,' he said with an expression of unironic sincerity. 'It might be that I offer my vision to the country, whether it's accepted or not. I think Russia virtually has to be rewritten from scratch.'

As things stand, it seems implausible that Kasparov should ever land the job of doing the rewrite. But it would be foolish to underestimate the boy from Baku. After all, he used to come up with ploys and gambits and moves that no one else in the world could see. And this particular game has only just started.