Paris and Marseille are held by the Socialists in local elections which saw the nationalist right win in Nice and certain towns.
The capital could shift to the right after 25 years under the Socialists if Rachida Dati defeats Emmanuel Grégoire.
Awkward choices emerge for mainstream parties after the first round of mayoral elections.
The highest-profile contest is for the mayorship of Paris - which has been under left-wing control for 25 years.
Eleven people are in detention in connection with the killing of far-right student activist Quentin Deranque.
Paris correspondent Hugh Schofield assesses the implications of a court ruling that 10 people are guilty of cyber-bullying Brigitte Macron.
The former French president was jailed for taking part in an election campaign funding conspiracy.
Sebastien Lecornu narrowly survived a crucial test on Tuesday as France's divided parliament voted in favour of a 2026 budget bill.
Sébastien Lecornu, the outgoing prime minister, said MPs recognise the pressing need to pass a budget by the end of the year.
