Polls show that leftist parties will gain the majority of 577 seats in the National Assembly, the lower house of parliament.   This means that the Socialist President Francois Hollande who took power last month would have the ability to enact measures he hopes will help curb unemployment and jump start Europe's second largest economy.

But it is not clear whether the Socialists and their allies will win the necessary majority in the first round against the conservative UMP party, led by former President Nicolas Sarkozy.

Early returns are expected when the polling stations close Sunday evening. A second and final round of voting is scheduled for June 17.

France's Senate, the upper house of parliament, is already under left-wing control.

Hollande unseated Sarkozy in a May 6 runoff election.  He needs lawmakers' support to persuade European Union leaders, especially German Chancellor Angela Merkel, to focus on spurring economic growth rather than enforcing austerity in struggling European economies.

France has about 45 million eligible voters.  Those living in remote overseas territories, such as French Guyana, French Polynesia and the Caribbean islands of Martinique and Guadeloupe, began voting Saturday.

Marine Le Pen's far-right National Front is seeking to win a voice in parliament for the first time since the 1980s after she placed third in the presidential race.