It is the match of his life. As correspondent Steve Kroft reports, the odds are long, and the dangers considerable, but Kasparov believes the soul of a nation is at stake.
Throughout his career, Kasparov intimidated opponents with his intensity, creativity, and daring. He could look at a chessboard and see 15 moves ahead. At a Toronto exhibition in June, he took on 20 opponents and beat all of them in an hour-and-a-half.
But his latest undertaking -- leading protests against a powerful and popular government -- is a much more difficult challenge. This time it is Kasparov who is considered the amateur, playing against grandmasters in the Kremlin.
Asked what the difference is between chess and politics, Kasparov tells Kroft, "Chess has rules. And everybody has to follow the rules. And in Russian politics there are no rules at all. Except one rule: the Kremlin, our opponents, changing rules at their convenience anytime they want."
And the man who changed them is President Vladimir Putin.
When Putin was elected seven years ago, regional governors were selected by the people; now they are appointed by Putin's office. Big media outlets once receptive to a wide range of political views have fallen under state control, and opposition groups like Kasparov's "United Civil Front" say they have been shut out of the national debate and cut off from the electoral process.
"We're facing a very dangerous regime that is threatening not only the future of my country but the stability of the whole world," Kasparov says.
What is he opposing?
"Every element of this regime. No elections. Censorship. The destruction of all democratic institutions," Kasparov says.
Kasparov doesn't believe that today's Russia is a democracy. He calls it a "police state."
When he has been able to organize protests, Kasparov has been met with thousands upon thousands of riot police and special troops brought in from the countryside to ensure order and discourage participation, sometimes with truncheons.
Kasparov has been detained, fined, and investigated. He assumes his political headquarters in Moscow is bugged and that he is under constant surveillance.
In December, the authorities raided his office looking for evidence Kasparov had violated a vaguely-worded extremism law, which makes it a crime to make false statements about government officials. It was enacted to fight terrorism, but now Kasparov says it is being used to stifle dissent. But it hasn't stopped him from speaking out.
"I would probably say that Putin doesn't run the country, he runs a corporation. Call it KGB Incorporated," Kasparov says. "He is working on behalf of the ruling elite that wants to benefit from looting the country."
Asked if he is an "extremist," Kasparov tells Kroft, "I'm afraid that by current Russian law, yes. Because that's what Russian law implies."
"Technically speaking," Kasparov says he could go to prison just for criticizing the government or staging a rally. And he says the penalties could be up to six years in prison.
It would not be the first time government critics have been silenced. Mikhail Khodorkovsky was once one of the richest men in Russia, until he displayed some political ambition. Now he's in a Siberian prison. The investigative reporter Anna Politkovskaya was shot to death in her Moscow apartment building. And former KGB agent Alexander Litvinenko was poisoned in London with polonium.
There's no evidence linking Putin to the deaths, but there is no denying that bad things tend to happen to people who challenge authority.
"There have been people in this country, prominent people, who have lost their lives. Does that concern you?" Kroft asks.
"It does," Kasparov admits. "You know I'm [a] normal human being. I have my private life. And my loved ones concern about my safety. I try to reduce the risk."
Asked if he thinks he's still protected somewhat by his fame and the fact he's still considered a national hero, Kasparov tells Kroft, "It does help to a certain degree. But it is not an ultimate protection. No one is safe in Putin's Russia."