Georgians are voting in snap presidential elections, in what is seen as a democracy test for the former Soviet republic.
President Mikhail Saakashvili called the poll, after suppressing huge opposition protests in November.
The pro-Western leader wants not only to be re-elected but to prove his democratic credentials to his critics.
Opposition parties have accused the authorities of trying to rig the vote. The government denies the charge.
Along with the presidential election, Georgians are being asked to vote on whether they should have a parliamentary election in the spring of 2008, and whether the country should join Nato.
Hundreds of foreign observers are monitoring the ballot.
Polling stations in the country of some five million people opened at 0800 (0400 GMT) and will close at 2000.
President Saakashvili has led the field of seven candidates in opinion polls, but it is not clear if he could win an outright majority to avoid a run-off.
The polls also suggest that Mr Saakashvili's closest rival will be Levan Gachechiladze, a wine businessman and independent MP chosen by the main opposition bloc as their candidate.
According to the BBC's Neil Arun in Tbilisi, heavy snow overnight gave the grey streets of the capital a bright, festive feel.
He says the mood is generally cheerful, with voters gathered at polling stations taking the opportunity to mingle with neighbours and friends.
At a polling station in Tbilisi, Nodar Zardiashvili said he had voted for Mr Saakashvili.
He told the AFP news agency that he backed the president "because he is doing the right thing by taking the country into Nato and the European Union".
Nino Saladze, another voter in the capital, said she was supporting Mr Gachechiladze.
"We've had enough of Mr Saakashvili, November was the last straw," she told AFP.
On the eve of the elections, Mr Saakashvili said Georgia was still a democratic pioneer among former Soviet republics, despite the crackdown on the opposition protests in November.
He said the move - which also included the imposition of a state of emergency - was unpleasant but necessary in order to prevent the government's violent overthrow.
"We have to show the whole world that Georgian democracy is still alive," he told thousands of supporters at a final campaign rally in the capital, Tbilisi.
But Mr Gachechiladze, who represents nine opposition groups, alleged "there are violations" in the poll, without giving further details.
"What is currently happening in Georgia is not a free election," he said earlier.
Mr Gachechiladze has also complained that "we cannot use media outlets or promotional means" and that a "smear campaign" was being staged against the opposition in the media.
Correspondents say Mr Saakashvili has run a well-funded election campaign, while other candidates have been much less visible.
Opposition groups have also accused Mr Saakashvili of authoritarian tendencies and a failure to tackle large-scale social deprivation in Georgia.
Mr Saakashvili, a US-educated lawyer, came to power after street protests in 2003, dubbed the Rose Revolution.
His first term as president has seen Georgia strengthen its ties with Nato and the European Union.
But relations with Moscow have soured and Georgia's economy has been badly hit by a Russian ban on Georgian goods.
Georgia's proximity to Iraq, Iran and Turkey makes it strategically important. A key oil export pipeline from the Caspian Sea also runs through it.
The West regards Georgia as a key test of Russia's readiness to respect other ex-Soviet states' independence, while Moscow is sensitive to any potential source of instability along its border in the Caucasus.