The hotly contested race has pitted ex-French ambassador and Georgia's former foreign minister Salome Zurabishvili, supported by the ruling Georgian Dream party, against opposition leader Grigol Vashadze, also a former foreign minister.
The two have an almost equal chance of being elected, according to opinion polls conducted in the run-up to the vote.
Vashadze -- backed by exiled ex-president Mikheil Saakashvili's United National Movement and 10 other opposition groups -- has been boosted by growing popular discontent over the government's failure to tackle poverty, AFP reported.
He has promised to mount a campaign demanding snap parliamentary polls, if elected.
Both candidates have campaigned on similar promises to bring Georgia closer to full membership of the European Union and NATO.
The vote is Georgia's last direct leadership poll as the Black Sea nation transitions to a parliamentary form of governance following a controversial constitutional reform.
The inauguration of a new president will pave the way for the new constitution to enter into force, making the head of state a largely ceremonial figure.
Georgia's next president will be elected in 2024 by a 300-member electoral college.
The constitutional change was adopted in September 2017, a move protested by all opposition parties, which denounced it as favoring the ruling party.
Georgia's outgoing President, Giorgi Margvelashvili, refused to run for a second term, saying he is not interested in assuming the reduced role.
Over 3.5 million people are eligible to vote in the election, which will be monitored by international observers from the Organization for Security and Cooperation in Europe.
Polls opened at 0400 GMT and will close at 1600 GMT, with results expected to start coming in overnight.
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