Country elects mayors for its largest cities
If, as polls suggest, no candidate wins more than 50 percent of the initial city votes, run-offs between those who come first and second will be held on June 19.
Only in Turin is the candidate of Renzi's Democratic Party (PD) a clear favourite. Renzi has said the elections will have no repercussions for his left-right coalition government.
Some 13 million people, or a quarter of the adult population, are eligible to vote for mayors of around 1,300 towns and cities, though political attention is focused firmly on the biggest, Rome, Milan, Naples and Turin.
Rome offers an opportunity for 5-Star to capitalise on anger towards the main parties on the left and right which have been embroiled in corruption scandals.
Five-Star appeals to voters across the political spectrum with a simple slogan: honesty.
It promises to introduce universal income support for the poor and to hold a referendum on whether Italy should remain in the euro. But its chief asset remains its image as the only party that is serious about taking on corruption and privilege.
Voting opened at 7 a.m. (0500 GMT) and will close at 11 p.m., when the result of exit polls will be announced for the main cities. Initial projections based on the actual vote count will be issued after around an hour, and then at regular intervals.
Five-Star's candidate Virginia Raggi, a 37-year-old lawyer, hopes to become Rome's first woman mayor and was ahead in opinion polls before their publication stopped 15 days before the vote, as required by Italian law.
Milan is probably the key contest for Renzi. His chosen candidate Giuseppe Sala, the former head of the Expo world fair held in the city last year, was expected to win comfortably but now looks seriously threatened by a moderate centre-right rival.
In Turin, incumbent mayor Piero Fassino, one of the PD's best-known national figures, was well ahead in the polls.
Naples is widely expected to be held by the leftist incumbent Luigi de Magistris, an independent former prosecutor who has declared Naples a "Renzi-free zone".
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