Security has been stepped up in Paris in the wake of the terror attacks, with Britons there advised by the Foreign Office to "exercise caution in public places".
While it had initially been reported that France had closed its borders, officials confirmed they were instead reinstating border checks.
Some passengers due to travel to Paris from the UK have been offered exchanges or free cancellations should they no longer wish to travel to the city.
Eurostar services are running as normal to Paris, with the first service to Paris Gare Du Nord from London departing as scheduled at 6.18am, though there were reports that many seats were empty.
The operator tweeted: "Our deepest sympathy and condolences to all those affected by the tragic events in Paris this evening."
They offered a free exchange for passengers who had planned to travel today but have decided not to, and directed those who plan to travel anyway to check their website for updates.
Eurotunnel services for cars and lorries from Folkestone to Calais were also running to schedule.
Easyjet said it is operating its flights to Paris as normal but anyone travelling in the next 48 hours can cancel and get a refund. Passengers are advised to contact Customer Services.
Another airline, Flybe, said any customer travelling today or tomorrow and who no longer wished to do so could rebook to a later date to any alternative destination, or obtain a credit note for future travel.
Ferry services from the port of Dover are open as usual, with some disruption to the Calais service due to weather, while the Dunkirk route is running as normal.
In Paris the Metro lines 3,5,8,9 and 11 are closed and bus services crossing the 10th and 11th arrondissements, where some of the attacks took place.
Tourists planning to go to Disneyland Paris were also being warned that the attraction was closed for the day in the wake of the attacks.
A statement on the attraction's website said: "In light of the recent tragic events in France and in support of our community and the victims of these horrendous attacks, Disneyland Paris has decided not to open its theme parks on Saturday 14 November."