Greeks are voting in crucial elections which could determine the country's future in the eurozone.
The main contenders, the right-wing New Democracy and left-wing Syriza, are at odds over whether broadly to stick with the tough EU bailout deal, or reject it and boost social spending.
Opinion polls are banned for two weeks before voting but unofficial polls say the result is too close to call.
EU leaders say if Greece rejects the bailout, it may have to leave the euro.
The election, the second in six weeks, was called after a vote on 6 May proved inconclusive.
"Today the Greek people speak. Tomorrow a new era for Greece begins," New Democracy leader Antonis Samaras said after casting his ballot.
"We have conquered fear. Today we open a road to hope," Syriza leader Alexis Tsipras said.
The election marked the start of a new era for Greece as "an equal member of a Europe that is changing", he added.
The election has so far passed off peacefully, apart from an incident at the prominent private media group Skai - its offices near the port of Piraeus were evacuated after two grenades were found outside, according to local media reports.
The grenades were found after an anonymous phone call to the group, whose outlets have backed the unpopular austerity drive.
Sunday's vote is being watched around the world, amid fears that a Greek exit from the euro could spread contagion to other eurozone members and send turmoil throughout the global economy.
Tough austerity measures were attached to the two international bailouts awarded to Greece, an initial package worth 110bn euros (£89bn; $138bn) in 2010, then a follow-up last year worth 130bn euros.
Many Greeks are unhappy with the conditions attached to deals which have been keeping Greece from bankruptcy. All but one of the parties standing for election have promised some degree of renegotiation of the terms.
"We want to believe that there will be an elected government on Monday, in order to see a brighter day for this country. Because we have gone through a terrible ordeal, both financially as well as emotionally and in every aspect there is," Athens voter Efrosyni Papageorgiou told the BBC.
"We are worried," said lawyer Pantelis Traianos.
"We truly wish for a new government to be elected and we wish to remain in the eurozone. On the other hand, the bailout was truly harsh, the measures were unbearable, there will be a need without a doubt for a re-negotiation."
In remarks quoted by the Reuters news agency a few hours before polls opened, the head of the Organisation for Economic Co-operation and Development Angel Gurria suggested that the next Greek government should be given a chance to revisit the bailout conditions.
"If that is the condition presented for Greece to stay [in the eurozone] and then move on, I would say it is probably something that should be attempted," he was quoted as saying.
But Germany, which has the eurozone's most powerful economy, insists Greece, like other member states which have received international bailouts, must abide by the austerity conditions.
On the eve of the vote, Chancellor Angela Merkel said: "It is extremely important that tomorrow's Greek elections lead to a result in which those who form the government say, 'Yes, we want to keep to our commitments.'"
Like Mr Gurria, the German chancellor and several other European leaders will be attending the G20 summit in the Mexican resort of Los Cabos on Monday, which is set to be dominated by the eurozone crisis and the aftermath of the election.
Greeks were celebrating hours before the polls opened, after their national football team qualified for the quarter finals of Euro 2012 with a surprise 1-0 win over Russia.
"We are proud that we gave the people back home some joy and a break from their problems - even for a short while," striker Georgios Samaras said.
The Kathimerini website noted that Greeks had few reasons to feel national pride at the moment, but sport had provided them with plenty of it. The victory could lead to a quarter-final tie against Germany.