The latest polls show a narrow victory for the center-right European People's Party, which includes national parties led by German Chancellor Angela Merkel and Spanish Prime Minister Mariano Rajoy. The EPP is expected to get 28 percent.
The center-left Progressive Alliance of Socialists and Democrats of French President Francois Hollande and Italian Prime Minister Matteo Renzi is likely to finish second with 26 percent.
Behind them will come Liberals, Greens, a group led by the British Conservatives and a growing assortment of radicals on the left and right who share a common dislike for the European Union.
But the overall winner could well be voter apathy.
Since elections to the EU's assembly were first held in 1979, turnout has fallen from 62 to 43 percent even as the parliament's power has increased.
Fewer still are predicted to show up for the current elections, which will be held from Thursday to Sunday, depending on the country. Less than 30 percent are expected to vote in some countries.
Although millions of Europeans don't care, the election does matter — including for people far beyond Europe's borders. Here's why:
1. The EU is huge.
Taken as a whole, the EU rivals the United States as the world's largest economic power. Its $16.2 trillion annual output remains well ahead of China’s.
The EU is the world's biggest exporter and is the leading trading partner for more than 80 countries, including China, Brazil and Russia. When the euro currency tottered close to collapse in 2011, it threatened to bring the world economy down with it.
The EU is the world's largest donor of development aid, and takes a leading role in setting global climate change rules and in international negotiations on issues from sanctions on Russia to Iran's nuclear program.
The European Parliament not only has a direct influence on EU policy but, in an innovation this year, the election should decide who will head the union's executive body.
In theory, the candidate of the winning party will become president of the European Commission — although the appointment will need to be agreed by national leaders from the 28 EU countries.
The frontrunners are Jean-Claude Juncker, a center-right former prime minister of Luxembourg, and German Social Democratic politician Martin Schultz.
2. The Parliament actually does useful stuff.
For years, the European Parliament was a toothless talking shop.
Members were viewed as an overpaid waste of taxpayers' money, their reputation further tarnished by frequent cases of expense fiddling and a costly shuttle between the assembly's three homes in Brussels, Luxembourg and the eastern French city of Strasbourg.
That shuttle remains, but the parliament now has genuine power. A succession of EU treaty revisions have greatly boosted its influence: It now acts like a real parliament with joint responsibility with national governments to block or amend laws and policies put forward by the executive commission.
Since the last elections, the parliament has used its powers to shape polices on issues ranging from maternity leave to banking regulation, carbon emissions to data protection and internet freedom. It also had a key role in approving measures to prevent a repeat of the euro crisis.
3. The elections could usher in radical changes in Europe's political landscape.
Like mid-terms in the United States, voters use European Parliament elections to punish incumbents. But while Americans have limited voting options, Europeans have a vast array of weird and often not-so-wonderful parties to choose from.
This week, record numbers are expected to pick radical new parties from the far left, ultranationalist right and any number of populist alternatives that claim to sit somewhere in between.
Among those expected to do well is France's National Front, whose former president Jean-Marie Le Pen on Tuesday said the ebola virus could help solve his country's immigration problem. Also Italy's Five-Star Movement, whose leader Beppe Grillo has promised online trials with his party members acting as a jury to convict wayward politicians, industrialists and journalists. And Greece’s Nazi-influenced Golden Dawn Party, whose leader Nikolaos Michaloliakos is in jail awaiting trial for running a criminal organization.
Although there’s a vast range of opinion among the various new movements, in general those on both the left and right flanks share a dislike of the EU itself. They could take a quarter of the seats.
In nine of the 28 countries, radical parties are challenging for first or second place.
Experience shows groups on the far right find it difficult to work together, but nationalists from France, the Netherlands, Austria, Italy, Belgium and elsewhere are hoping to form a united faction in the parliament that could disrupt EU business.
Although voters often revert to mainstream parties in national elections, their time in the European Parliament can help give the rabble-rousers a platform to build support and establish their public profiles. The far right's poll success is already influencing mainstream parties to take harder lines on issues such as immigration and the EU’s role.
4. The vote may give a boost to Putin and deal a blow to trans-Atlantic trade.
Europe's far right loves Russian President Vladimir Putin and has been supportive of his work in Ukraine.
"Mr. Putin is a patriot," National Front leader Marine Le Pen (daughter of party founder Jean-Marie) said last week. "He's aware that we defend common values, the values of European civilization," she told Austria's Kurier newspaper.
Heinz Christian Strache, the leader of Austria's Freedom Party, called Putin a "pure democrat."
Hungary's Jobbik Party has hailed the annexation of Crimea as exemplary. Hungarian prosecutors this week reportedly requested the European Parliament lift the immunity Jobbik representative Bela Kovac enjoys as a member so he can face charges of spying for Russia.
With Europe already struggling to agree on a united response to the crisis in Ukraine, the arrival of a hard-line pro-Putin caucus in the parliament could make that task even harder.
Meanwhile, the new parliament could hold back efforts to strike a huge trans-Atlantic free-trade deal as parties on both the far left and far right are opposed to the Transatlantic Trade and Investment Partnership.
While supporters claim a deal could boost the world economy by $425 billion, the TTIP's many opponents fear it will undermine European labor and health standards, increase economic dependence and destroy jobs.
Last year, the parliament voted overwhelmingly to open negotiations. The new legislature, which will have to approve any deal, will be wary of Europe granting too many concessions to Washington.