The crackdown follows multiple attacks on bars, restaurants, a concert hall and a stadium in Paris on Friday, in which 129 people were killed.
A police operation is also under way in Brussels, Belgium. Shots and explosions have been heard in Molenbeek district.
Meanwhile the brother of two suspects, arrested in Belgium, has been freed.
Mohammed Abdeslam's lawyer said on Monday he had been released without charge.
His brothers are Brahim Abdeslam, killed during the attacks, and Salah, who is on the run.
Five out of seven people arrested in Belgium at the weekend have been released, AFP news agency reports.
France held a nationwide minute of silence at midday local time (11:00 GMT) for the victims.
French PM Manuel Valls said the attacks had been organised from Syria.
He added that the authorities believed new terror attacks were being planned in France and other European countries.
Meanwhile two more Paris attackers were named, along with five already identified. One is confirmed to have entered Greece as a migrant earlier this year.
Investigators are also reported to be focusing on a Belgian of Moroccan descent who is described as the possible mastermind of the attacks.
Abdelhamid Abaoud, 27, lived in the same neighbourhood of Brussels as two of the attackers and is now believed to be based in Syria, where he has risen through the ranks of IS.
Police have named Brussels-born Salah Abdeslam, 26, as a key suspect, and a manhunt is under way. He was reportedly stopped by officers in the wake of the attacks while crossing into Belgium but then let go.
The BBC's Christian Fraser in Brussels says the Belgian authorities are confident that he is in the Brussels area.
Meanwhile, French aircraft have attacked Raqqa, the stronghold in Syria of the Islamic State group, which has said it carried out the attacks.
IS has issued a statement saying the raid targeted empty locations and that there were no casualties.
Mr Valls said France was dealing with a "terrorist army", rather than a single terrorist group.
"We know that operations were being prepared and are still being prepared, not only against France but other European countries too," he said.
Suspected Paris attackers
- Salah Abdeslam, 26 - urgently sought by police
- Brahim Abdeslam, 31 - named as attacker who died near Bataclan concert hall
- Omar Ismail Mostefai, 29, from near Paris - died in attack on Bataclan
- Bilal Hadfi, 20 - named as attacker who died at Stade de France
- Ahmad al-Mohammad, 25, from Idlib, Syria - died at Stade de France (unverified)
- Samy Amimour, 28, from near Paris - suicide bomber at Bataclan
- Two other attackers died during the assaults in the city
The prime minister said more than 150 raids on militant targets had been carried out in different areas of France early on Monday.
"We are making use of the legal framework of the state of emergency to question people who are part of the radical jihadist movement... and all those who advocate hate of the republic," he said.
Police sources told news agencies that properties in the Paris suburb of Bobigny, as well as the cities of Grenoble, Toulouse and Lyon, had been targeted.
Interior Minister Bernard Cazeneuve said 23 people had been arrested and dozens of weapons seized, including a Kalashnikov assault rifle and rocket launchers. More than 100 people have been placed under house arrest.
Seven attackers died in the assault on the French capital, most of them after detonating suicide belts.
Five were identified over the weekend. On Monday another two were named by the Paris prosecutor as Ahmad al-Mohammad and Samy Amimour.
- Al-Mohammad is the name on a Syrian passport found with the remains of one of the attackers, though the man's identity has not yet been verified. What has been confirmed is that his fingerprints match those taken by the Greek authorities after he arrived with migrants on the island of Leros in October 2015
- Amimour was said to be facing terrorism charges in France. He was placed under judicial supervision while under investigation for terrorist conspiracy - he planned to go to Yemen. An international arrest warrant was issued against him when he broke bail in autumn 2013. Three of his relatives were among those detained this morning
One of the main lines of investigation concerns Molenbeek, which has a reputation as being a haven for jihadists.
France is currently marking a second day of national mourning. A state of emergency declared by President Hollande remains in force. Thousands of extra police and troops are on the streets of Paris.
Main attack sites:
- Bataclan concert venue, 50 Boulevard Voltaire, 11th district - 89 dead when stormed by gunmen, three of whom were killed; another gunman died nearby
- La Belle Equipe, 92 rue de Charonne, 11th district - 19 dead in gun attacks
- Le Carillon bar and Le Petit Cambodge restaurant at rue Alibert, 10th district - 15 dead in gun attacks
- La Casa Nostra restaurant, 92 rue de la Fontaine au Roi, 11th district - five dead in gun attacks
- Stade de France, St Denis, just north of Paris - three attackers and a bystander killed