A source close to the investigation told AFP the attacks were coordinated acts of "sabotage".
"This is a massive attack on a large scale to paralyse the TGV network," SNCF told AFP, adding that many routes will have to be cancelled.
"SNCF was the victim of several simultaneous malicious acts overnight," the national train operator said, adding that the attacks affected its Atlantic, northern and eastern lines.
"Arson attacks were started to damage our facilities," it said, adding that traffic on the affected lines was "heavily disrupted" and the situation would last through the weekend as repairs are conducted.
Trains were being diverted to different tracks "but we will have to cancel a large number of them", the statement said.
The southeastern line was not affected as "a malicious act was foiled".
SNCF urged passengers to postpone their trips and stay away from train stations.
The attacks were launched as Paris was under heavy security ahead of the opening ceremony of the Summer Olympics, with 300,000 spectators and an audience of VIPs expected at the event.
The parade on Friday evening will see up to 7,500 competitors travel down a six-kilometre (four-mile) stretch of the river Seine on a flotilla of 85 boats.
It will be the first time a Summer Olympics has opened outside the main athletics stadium, a decision fraught with danger at a time when France is on its highest alert for terror attacks.
At Paris's Montparnasse train station, dozens of passengers were waiting for more information about their trips after delays of 30 minutes to almost two hours were announced.
"Normal traffic is expected to resume on Monday, July 29," read one of the signs in the departure hall.
"We arrived around 7:00 am but we were told that we might not be able to leave before Monday," said 27-year-old student Jocelyn, who had planned to travel to Bretagne and refused to give her full name.
"We expected it to be a bit chaotic in Paris with the opening ceremony scheduled for this evening, but we didn't think it could be this bad," she said.