One obvious front-runner is , Prime Minister and the man who stands in for Mr Yeltsin when the latter is incapacitated by alleged alcohol abuse, heart attacks, or both.
Mr Chernomyrdin formed a political party called Our Home is Russia, which was soon dubbed the party of government, because of the preponderance in its ranks of government apparatchiks and ex-apparatchiks.
Mr Chernomyrdin himself used to be boss of the giant Soviet gas and minerals conglomerate, Gazprom. He initially expressed strong reservations about the market economy ("a bazaar"), but gained international brownie points by demonstrating a commitment to keeping economic reform on track.
During the hostage crisis in Budyonnovsk earlier this year, he impressed Russians and international observers alike with his unexpectedly cool decisiveness in difficult circumstances.
Not a man of vision, but perhaps a safe pair of hands.
Alexander Lebed, a former Soviet general, is an unpredictable figure who seems to believe he is the man to save Russia. His politics are unclear, but his core support comes from those who feel that a silnaya ruka, or strong hand is needed, to run the country. - a potentially large constituency.
He criticised Russia's war in Chechnya, but not because he was soft on the Chechens. His criticism appeared to be that the war had been fought badly, rather than the fact that it was fought at all.
Until recently, he was commander of the Russian troops in the former Soviet republic of Moldova, where ethnic Russians and Moldovans (Romanians) have fought a low-key war in recent years.
Yuri Skokov, General Lebed's fellow-leader of the Congress of Russian Communities, is seen by some as his future rival. President Yeltsin himself has been keen to stoke such speculation in recent weeks - not least, perhaps, in order to put General Lebed and Mr Skokov at each other's throats, and thus divide their support.
The pathologically self-confident Vladimir Zhirinovsky has always been flamboyant, with his extremist nationalism. Yesterday, in typical form, he appeared to blame Mr Yeltsin's illness on the machinations of President Clinton and the CIA.
Mr Zhirinovsky's neo-fascist Liberal Democrats gained a large slice of the vote at the last parliamentary elections, in 1993. His recipe of Make Russia Great Again went down well with the embittered voters, who were happy to blame Jews and assorted foreigners for all Russia's ills.
There is plenty of bitterness still around. But Mr Zhirinovsky has begun to be perceived as a kloun, even by his former supporters
One possible alternative for the discontented might be the Communist Party leader, Gennady Zyuganov. Communist support has grown in the past year.
One of the liberals' few remaining hopes is Grigory Yavlinsky, who came to prominence in the dying days of the Soviet Union, when he created a radical economic plan in 1990 which was rejected by Mr Gorbachev. His party, Yabloko, - an acronym for the "Yavlinsky bloc'' -receives the support of some of the tiny band of liberal Russian voters. One of the other liberals is the former prime minister Yegor Gaidar, who was the main architect of Russia's economic reform programme, and became very unpopular as a result.
At the moment, Mr Yavlinsky's chances look poor. But it is perhaps worth noting that predictions about Russian politics have a poor track record. Disillusion is so great that "a plague on all their houses" tends to be the commonest voters' response. As a result, sure-fire winners can turn out to perform badly, and no-hopers can easily come in first.
That, of course, includes Mr Yeltsin himself, the man who has been written off more often than any other politician in the entire Russian Federation.
A potential candidate who is more highly regarded in the West than in Russia itself is Mikhail Gorbachev. The former international superstar, who is now almost entirely ignored by his compatriots, runs the Gorbachev Foundation, and - like his old friend, Baroness Thatcher - spends much of his time on the international lecture and chat-show circuit.
At home, the former Soviet leader is still distrusted by the liberals - who feel that he put the brake on radical reform, as opposed to Communist perestroika - and hated by the hardliners - who argue that he destroyed the wonderful and flourishing Soviet Union. But there is less open hostility to him than there was a year or two ago. At least he has shown himself to be (a) honest and (b) not a lunatic, which is more than can be said for many Russian politicians today.