The frontrunner among the 23 candidates is Giorgi Margvelashvili – a close ally of PM Bidzina Ivanishvili, Mikheil Saakashvili’s bitter rival.
Following the polls, constitutional amendments will significantly reduce presidential powers.
Mikheil Saakashvili is barred from standing as he has already served two terms.
Bidzina Ivanishvili’s win in parliamentary elections last year ushered in the former Soviet republic’s first legal transfer of power.
The billionaire businessman also intends to resign within weeks of the vote, saying he has achieved his aims.
He has called Sunday’s vote Georgia’s “first European-style election”.
Polling stations are open from 08:00 local time until 20:00. Final results must be published no later than November 16.
If no candidate wins more than 50% of the vote, a run-off between the two top-placed candidates must be held two weeks after the results of the first round are published.
Opinion polls suggest a win for philosopher and former Education Minister Giorgi Margvelashvili, a member of Bidzina Ivanishvili’s Georgian Dream coalition.
But a second round with David Bakratze, who represents Mikheil Saakashvili’s United National Movement (UNM) remains a possibility.
Nino Burjenadze, another top contender, is one the most well known female politicians in Georgia.
Georgia is moving closer to a parliamentary democracy although the president will formally remain the head of state and retain an important role in foreign policy, our correspondent adds.
Since coming to power in the bloodless 2003 “Rose Revolution” Mikheil Saakashvili has implemented reforms which helped root out corruption.
He pursued a pro-Western foreign policy and improved public services in the Caucasus republic, where poverty remains widespread.
But the country’s prime minister, who is also Georgia’s richest man, has dismissed these achievements, calling Mikheil Saakashvili a “liar” and a “dictator”.
Bidzina Ivanishvili has indicated that the outgoing president could face questioning or even prosecution over the alleged crimes of his government after his term ends. He has not yet named a successor as prime minister.
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