Two of the men arrested in the southern town of Lunel had returned from Syria, according to French police sources.
The five Russian citizens are reportedly connected to organised crime and are not believed to have been planning a terrorist operation.
The arrests of people suspected of giving logistical support to the three terrorists who claimed 17 lives last week were in suburbs around Paris.
The arrests for 'conspiracy to prepare acts of terrorism' focussed on a network allegedly based in the south-west city of Toulouse.
A total of 13 people, including the presidents of Marseille and Caen football clubs, were arrested Tuesday in separate corruption probes.
Two minors were among those arrested near Lyon on suspicion of recruiting young women to fight in Syria.
The pair, aged 15 and 17, were arrested on suspicion of helping a 14 year-old girl travel to Syria to join jihadist forces.
Cars were set alight and rocks thrown at police, who reported 14 officers injured, in clashes after the outlawed protest at Israel's Gaza offensive.
Algerians in France celebrated their team's achievement in reaching knockout stages; there were 74 arrests as some fans clashed with police.
The son-in-law of wealthy property owner Hélène Pastor is also among those being questioned following her ambush murder in May in Nice.
The four were detained in Marseille over allegations that they have been helping to recruit volunteers for radical Islamist networks in Syria.
Six people have been arrested for attacks on individuals performing the 'quenelle', a gesture associated with anti-Semitism, on social networks.
Veterinarians and other meat industry workers are suspected of selling for human consumption meat from horses used for medical research.
The 45 people arrested in France and in French overseas territories were detained as part of an operation targeting international traffickers.
French investigators unraveling the harrowing case of an infant girl found in the boot of a car believe she may have been hidden there since birth.