The focus is on the major cities of Bologna, Milan, Naples, Turin, and especially the capital Rome, where the populist anti-establishment Five Star is heading the race for the mayor's seat.
More than 13 million people are eligible to choose members of 1,300 municipal councils in a two-round ballot to be completed on June 19th.
By midday, turnout stood at almost 18%, the interior ministry said.
Despite the possibility his center-left Democratic Party (PD) could lose control of both Rome and Milan, Renzi has played down the significance of the vote.
On Sunday, as if to demonstrate his insouciance, the 41-year-old prime minister visited the training camp of the Italian football squad, to wish them luck in the Euro 2016 championships starting in France next week.
“The municipals are about mayors, the people whose job it is to repair the streets, not the government of the country,” Renzi said recently.
Rome has been without an elected leader since last October, when Ignazio Marino, a member of Renzi's PD, was forced to quit over an expenses scandal.
Voting in the Trastevere district of the capital, Rolando Antonucci, 78, told AFPTV he hoped for greater civic-mindedness “among citizens and lawmakers alike.”
“I came to vote because I have hope… who knows, maybe for once I will be satisfied with my vote.”
By 1700 GMT, turnout stood at just over 46 percent, the interior ministry said.
Rome is saddled with a debt of more than €13 billion ($15 billion), which is twice its annual budget.
The expenses scandal and a much bigger unrelated scandal over organized crime's infiltration of Rome's City Hall have bolstered Five Star's Virginia Raggi.
The 37-year-old went into Sunday's vote with polls indicating she could secure around 30% of first-round votes. The PD's Roberto Giachetti was trailing on around 24%.
Losing Rome would not augur well for Renzi four months before a referendum on constitutional reforms designed to end decades of gridlock in parliament.
Renzi has vowed to resign if voters reject the reforms.
In Milan, the PD candidate is locked in a tight race with the candidate of the right.
Five Star meanwhile is hoping that success in Rome will give it the platform it needs to transform itself into Italy's main opposition in the run-up to national elections due by June 2018 at the latest.
With former premier Silvio Berlusconi now a fading figure on the national stage, Italy's right is being reshaped and the battle for its leadership is being played out in the capital.
Giorgia Meloni, a candidate put up by one of several small groups that emerged from Italy's neo-fascist movement, is being backed by the anti-immigrant Northern League, whose leader Matteo Salvini wants to unite all the right behind himself.
But Berlusconi has backed another candidate Alfio Marchini, having told the pregnant Meloni that the role of mayor was not compatible with motherhood.
There are concerns turnout could be low, with millions of Italians enjoying a long holiday weekend as a result of Republic Day falling on Thursday.
In a referendum in April on offshore oil platforms, the turnout was a meagre 32%