In total, 400 million people are eligible to take part. Voting has already taken place in seven of the EU's 28 member states.
Results will be released after polls close at 21:00 GMT (22:00 BST) on Sunday.
In Britain the focus will be on how pro- and anti-Brexit parties perform.
UK election law prevents the BBC, along with other UK broadcasters, from reporting details of campaigning or predictions of results until all polls have closed.
Across Europe, right-wing nationalist parties have mounted a co-ordinated challenge to the centre-right and centre-left groupings that have been the dominant power in every European Parliament.
Italy's League party has joined forces with other right-wing groups including Germany's AfD, the Finns Party and the Danish People's Party to create the European Alliance for People and Nations.
In doing so they hope to change the balance of power in the EU.
Although nationalist and Eurosceptic groups are currently among the smallest in the European Parliament, support has grown for anti-immigration parties since the 2014 election.
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