Left-leaning Moon Jae-in is the clear front-runner with centrist Ahn Cheol-soo his nearest challenger.
The election is being closely watched at a time of heightened tensions with North Korea.
Mr Moon wants to increase contact with the North - in contrast to Ms Park who cut almost all ties to Pyongyang.
BBC Korea correspondent Stephen Evans says this could lead to tensions with Washington.
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On the final day of campaigning, Mr Moon called for an "overwhelming victory", saying it would help to heal a national divide caused by the downfall and impeachment of conservative Ms Park.
Mr Ahn, campaigning in swing states, said he was still confident he could win the election.
In March, Ms Park became the first South Korean president to be removed from power by impeachment and is due to stand trial on charges of bribery and abuse of power. She has denied any wrongdoing.
Mr Moon narrowly lost to Ms Park in the last presidential election in 2012.
He has been critical of the two previous conservative administrations for failing to stop North Korea's weapons development.
He says he favours a two-track approach to encourage change, using increased dialogue alongside sanctions.
Our correspondent says that if US President Donald Trump pursues a hard line against the North - especially if it stages another nuclear test - Mr Moon's approach could cause greater tension between Seoul and Washington.
As well as North Korea, domestic issues are high on the agenda for voters. All the candidates are promising to protect the fragile recovery in South Korea's economy and to bring down youth unemployment, which remains stubbornly high.
[BBC SOURCES]
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