Saakashvili has led the field of seven candidates in opinion polls. His challenge is to secure more than 50 percent of the vote to avoid a runoff in two weeks' time.
Opinion polls suggest that Saakashvili's closest rival is Levan Gachechiladze, who represents a coalition of opposition groups.
Polling stations in the country of some 5 million people closed at 1600 GMT. Partial results are due to be published during the night.
In Tbilisi, RFE/RL Georgian Service Director David Kakabadze said there were reports of irregularities. "Technical violations [were reported] in some of the regions -- certain polling stations were opened later than planned. And our correspondent in one region reported that in one polling station the [local election] commission rejected objections from observers."
Kakabadze also said bad weather would affect turnout. "It's [been] snowing continuously for the past two days, so it's actually very difficult for people to get to the polling stations, especially in remote parts of Georgia. Even in Tbilisi people have some problems."
Saakashvili cut short his five-year term and ordered the early poll after police violently dispersed antigovernment protesters in November 2007. The president also imposed a state of emergency that included bans on independent television news broadcasts.
The violent crackdown angered many Georgians and called into question Saakashvili's commitment to democracy. Hundreds of foreign observers have come to Georgia to monitor the vote.
Burul Usmanalieva, a member of the Kyrgyz nongovernmental organization Coalition for Democracy and Civil Society, was in western Georgia to monitor the vote. She told RFE/RL's Kyrgyz Service that last year's government crackdown angered many Georgians.
"When I asked them, most of the youth wanted to elect Mikheil Saakashvili, but the older people are against him. They said there was a [protest movement] in November there. During the rallies [government forces] dispersed the crowd [of protesters], [who] were beaten up with truncheons. They are recalling it, saying: 'We cannot forgive it. He did a very bad thing, wasting the good job he had done within the last four years.' They are talking like that."
Usmanalieva said many people also complained that they were not on the voters lists.
Saakashvili, a U.S.-educated lawyer, came to power in a landslide election victory in 2004 after leading street protests known as the Rose Revolution in late 2003.
Speaking after voting today in Tbilisi, Saakashvili said that every citizen must feel that he is deciding the future of Georgian statehood. "With these elections today Georgia's fate is indeed being decided. These are not regular elections. So I want to tell all my fellow citizens that no matter how bad the weather is, no matter how many other problems you might have, I cordially ask and call upon you to take an active role in deciding our country’s fate."
Reforms initiated by Saakashvilis' government have attracted foreign investment and resulted in economic growth of between 9 and 12 percent annually. However, inflation has eroded incomes, utility bills have soared, and unemployment remains high.
Saakashvili warned today that the vote will also decide the course of Georgia's economic reforms. "This is [an] election where a lot of things are at stake: continuation of reforms, further advancement of the country. Georgia is a great success story for this region and now it is up to the people of Georgia to decide wether this success will continue as it was going until now."
Meanwhile opposition candidate Gachechiladze said today the opposition is wary of irregularities at the polls. "Our mood is very good and we are ready for the fight. I voted for ‘Georgia First And Foremost’ [Gachechiladze's campaign slogan]. These elections are marked with high activity, there are also of course violations but our headquarters is ready to react to all kinds of violations."
The opposition has accused Saakashvili of authoritarian tendencies and a failure to tackle large-scale social deprivation in Georgia. It complained against difficult access to the media during the election campaign.
The International Election Observation Mission deployed to monitor the election is to announce its preliminary conclusions on January 6. The mission is a joint undertaking of the Organization For Security and Cooperation In Europe (OSCE), the Parliamentary Assembly of the Council of Europe (PACE), and the European Parliament.
The mission's leader, U.S. Representative Alcee Hastings (Democrat, Florida), said the fairness of the election will show how committed Georgian authorities are to democracy. "Georgia has made commitment and expressed desires to become a part of the Euro-Atlantic family," he said. "They deserve that opportunity. This election is a watershed election that will make a determination as to their commitment."
Saakashvili's first term as president has seen Georgia strengthen its ties with NATO and the European Union amid animosity from Moscow.
Along with the presidential election, Georgians were asked in a nonbinding plebiscite to vote on whether the country should join NATO, and whether they should have parliamentary elections in the spring or fall of 2008.
(RFE/RL’s Georgian and Kyrgyz services contributed to this report.)
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