The vote is being held over four days
Voting for the European Parliament, the EU's most powerful legislative body, is under way, with the Netherlands and the UK the first to go to the polls.
Some 375 million people in 27 member states are eligible to vote. Most will cast their ballots over the weekend and results are expected soon afterwards.
Observers will be watching to see if turnout is higher than in 2004, when only 45% exercised their right to vote.
In the UK, elections are also being held in some areas for local councils.
The results of both polls are keenly awaited to see how they might affect the national political scene, following weeks of turmoil over MPs' expenses claims.
Correspondents say voters disillusioned with mainstream politicians may stay away altogether in disgust, or perhaps back smaller parties.
The vote is seen as a crucial test for the governing Labour Party, with a general election due within a year. Two senior ministers resigned earlier this week.
Irish voters go to the polls on Friday. Latvia, Cyprus, Malta and Slovakia vote on Saturday, while the Czech Republic and Italy vote over Friday and Saturday, and Saturday and Sunday respectively. People in the remaining 18 member states will vote on Sunday.
The Dutch government will give preliminary results from the vote on Thursday, but other countries will wait until Sunday evening.
Voters will decide who is elected to the 736 seats up for grabs under various forms of proportional representation. UK voters are electing 72 MEPs and the Dutch 25 on Thursday.
EU PARLIAMENT ELECTIONS
Thursday: UK and Netherlands
Friday: Ireland, Czech Republic
Saturday: Latvia, Cyprus, Malta, Slovakia, Italy and Czech Republic
Sunday: Austria, Belgium, Bulgaria, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Italy, Lithuania, Luxembourg, Poland, Portugal, Romania, Slovenia, Spain and Sweden
Results from 2000 GMT Sunday
Germany, with the biggest population, gets the most seats - 99 - while Malta will have just five.
The newly-elected MEPs will take their seats in one of Europe's most powerful political institutions, scrutinising and shaping legislation that covers everything from air quality to the cost of mobile phone calls.
Despite all this, in the past Europeans have failed to be moved by the elections, our correspondent says.
Last time around in 2004, turnout in the UK was 38% and in the Netherlands 39%. Some of the lowest turnouts were recorded in the newer member states of Central and Eastern Europe.
"I understand that people can feel tired of politics," said Czech Prime Minister Jan Fischer, whose country holds the EU's rotating presidency, on Wednesday.
"We often hear about a democratic deficit in the EU," he added. "One cannot complain of the EU being undemocratic and at the same time refuse to go to the polls," he said.
The BBC's Dominic Hughes in Brussels says this is the first big electoral test across Europe since the region was hit by economic meltdown.
And with voters tending to reflect on national rather than pan-European issues, the results could make uncomfortable reading for some governments, our correspondent says.
In the Netherlands, the Freedom Party (PVV) of the right-wing MP Geert Wilders is attempting to leapfrog the ruling Christian Democrats (CDA) and Labour Party (PvdA) and become the country's main representative in the European Parliament.
Geert Wilders says he will not take up his seat if elected as an MEP
Mr Wilders was refused entry to the UK in February on the grounds that he had sought to incite hatred with a film he made last year that equated Islam with violence and likened the Koran to Mein Kampf.
The controversial politician is also facing prosecution in the Netherlands for making anti-Islamic statements, following a court ruling the previous month.
Despite the charges many Dutch voters seem to like what Mr Wilders is saying, the BBC's Geraldine Coughlan in The Hague reports.
His Freedom Party is campaigning under the motto "For the Netherlands" and is extremely anti-EU, our correspondent says. Mr Wilders has said he will not take up his seat if elected as an MEP.
Polls show that Euroscepticism among Dutch voters has increased since the last European elections, with EU enlargement and integration the most unpopular issues, our correspondent adds.
One opinion poll suggests the Freedom Party could get as much as 12% of the vote, putting it level with Labour and only two percentage points behind the CDA.
Across Europe, far-right parties are hoping to win at least 15 seats. However, the centre-right European People's Party bloc is expected to remain the main force, followed by the European Socialists.