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
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.
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:
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Brussels remained under extremely heavy security Monday after police carried out raids around the Belgian capital amid warnings of possible terrorist attacks similar to those that killed 130 people in Paris 10 days ago.
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