French President Emmanuel Macron has called for "calm and order" after a major day of protests turned into rioting and looting on Saturday, and as some disorder continued in Paris on Sunday.

In a national address from the Élysée Palace, he warned those who had taken part not to "escalate", adding: "We all feel anger and shame."

Some 550,000 demonstrators took to the streets for largely peaceful protests against rising fuel prices, the cost of living and deteriorating public services, including schools.

Macron said a distinction must be drawn between the "legitimate demands" of protesters and "those who destroyed, smashed and looted".

Police have arrested about 2,400 people, according to France's interior ministry.

The French president said "nothing justifies" the "unbridled" violence seen and said those involved would face justice.

Groups of rioters, some dressed in black and wearing masks and balaclavas, also came out in Paris, Marseille and other towns and cities, with some setting fire to cars and buildings.

On Sunday, looting and clashes with police continued on a smaller scale in parts of the capital and Lyon.

Tear gas and stun grenades were fired at hundreds of people in Paris, according to the AFP news agency, and bottles and rocks were again hurled at police.

Sunday saw cities across the country confront the impact of what developed into one of the worst bouts of disorder in modern French history.

In Limoges, the ornate 19th Century city hall was gutted by a fire started by rioters on Saturday, its clock tower and roof collapsing and the flames took hold.

Speaking near the ruined building, Annèyse said: "This is very upsetting. Harming heritage buildings just because you are angry at the government is selfish, I think. It's hurting everybody."

"It's awful and unbelievable," said another man who gave his name as Gerrard. "We are very sad and I think it will be a long time to put it back as it was before."

Shopkeepers were seen early on Sunday morning picking up large shards of glass in Paris as rubbish trucks cleared up debris on some of the larger streets.

Widespread looting took place across the cities of Paris, Marseille and Lille, as well as other areas.

Macron said local prefects and mayors would draft measures to protections in place for shops and civic buildings.

The president also said there would be "full transparency" over the police response to the demonstrations.

Tear gas and water cannon were turned on crowds over the weekend and riot police were seen hitting people with batons in several places.

France's interior ministry said about 1,000 police officers were injured on Saturday, seven of them seriously.