The people of Poland are voting in a run-off election to determine the country's next president.
Observers say the final result between Bronislaw Komorowski and Jaroslaw Kaczynski could be close.
Mr Komorowski has been acting president since his predecessor, Mr Kaczynski's twin brother Lech, was killed in a plane crash as he travelled to Russia in April.
Results are expected to be released on Monday.
The election has been dominated by the crash. Poland's first couple and 94 other people died when their plane came down in Smolensk on 10 April as they flew to attend a memorial ceremony for the World War II Katyn massacre.
Clinton's supportThe contests pits the business-oriented Bronislaw Komorowski from the ruling centrist Civic Platform against the conservative Jaroslaw Kaczynski of the main opposition Law and Justice party.
Mr Komorowski won 41.5% in the first round and Mr Kaczynski 36.5%. Eight other candidates were eliminated.
Final opinion polls before Sunday's voting got under way suggested a close call, with floating voters certain to be pivotal.
Komorowski's support ranged from 45 to 54%, and Kaczynski's from 42 to 45%, AFP news agency reports.
Mr Komorowski's supporters see him working well with Prime Minister Donald Tusk in trying to curb fiscal deficit.
Mr Komorowski, 58, said at his rally on Friday: "You have the choice between a politician who harbours resentment and grudges, and the future, with an optimistic vision of Poland".
Mr Kaczynski's supporters hope a victory for their candidate would curb cuts in public spending.
He said on Friday: "This election began with a tragedy. I hope it will end with a great success for Poland".
Mr Kaczynski, 61, served as prime minister but was voted out of office in 2007.
The BBC's Adam Easton in Warsaw says Mr Kaczynski has tried to present himself as a changed man, prepared to compromise with opponents, in the wake of his own personal loss.
The Polish president has a role in foreign affairs and the power to propose and veto legislation, although policy is set by the prime minister.
US Secretary of State Hillary Clinton avoided taking sides during her visit to Poland on Saturday.
"Whichever candidate is chosen, the United States will continue to be your friend and partner," she said.
Polling stations close at 2000 (1800 GMT).
