About 50,000 police and 7,000 soldiers are being deployed around the country after Thursday's attack by a gunman.
Eleven candidates are vying to be the country's next president, with leading candidates spanning the political spectrum from far-left to far-right.
The two with the most votes will go to run-off round in a fortnight's time.
Four candidates are currently seen as being within reach of the presidency: the conservative François Fillon, the far-right leader Marine Le Pen, liberal centrist Emmanuel Macron and the far-left's Jean-Luc Mélenchon.