Eurostar has resumed operation of high-speed trains through the Channel Tunnel after three days with no service.
The company said all trains setting off on Tuesday had arrived as expected, and there were no reports of problems.
BBC correspondents reported passenger backlogs in both France and the UK were clearing, and those with on-the-day tickets are being allowed to travel.
The company blamed condensation on electrical systems for a series of breakdowns on its trains.
Meanwhile, Eurotunnel has also started to clear the backlog on its car shuttle service through the tunnel, having closed its services to new passengers on Monday.
Eurostar finance director Ian Nunn, speaking from London's St Pancras station, said: "We do have trains moving now and we are getting people on to trains, so that's a big relief for us but most importantly for them.
"We're terribly sorry for the discomfort we've put them through over the last few days and it's great to see that we're being able to make some inroads into the backlog and get people home."
'So angry'
Eurostar said it had space for 26,000 passengers to travel on Tuesday.
The BBC's Ben Georghegan at the Eurostar terminal in St Pancras said the passengers with on-the-day tickets - as well as passengers with pre-booked tickets - were being allowed to board.
Eurostar had prevented customers with on-the-day tickets from boarding while it cleared its backlog.
The first train left Paris at 0810 local time (0710 GMT), and the first departure from St Pancras was at 0740 GMT.
By late morning on Tuesday, four Eurostar trains had departed from St Pancras, three from Paris and two from Brussels. Each had a capacity of 750 passengers.
But some passengers were still angry with Eurostar after delays that began after trains broke down in the Channel Tunnel.
Stephanie Budd, 34, from Wallington in Surrey, finally arrived at St Pancras after taking her four children to Disneyland Paris.
She said: "The lack of information has been awful. Since Friday we have not known when we would be able to get home.
"It just was not explained that this would take so long. I'm so angry."
Train breakdowns
The problems began on Friday, when five Eurostar trains broke down in the tunnel after condensation affected electrical systems.
Another train, which had been laid on to try to clear the backlog, suffered the same fate as it left the tunnel in Kent on Saturday nightThe company said snow shields used to protect the trains' electrics had worked for the past 15 years, but a recent cold snap in France had been "unprecedented".
Eurostar trains were taken out of service while it tried to rectify the problem, causing a massive backlog of stranded passengers on both sides of the Channel.
The company tested modifications to trains on Monday, and said it was confident there would be no problems.
The problems with Eurostar had a knock-on effect for Eurotunnel train services carrying cars and lorries through the tunnel.
It closed its car shuttle service to new passengers on Monday after saying its terminal in Folkestone, Kent, had reached "saturation" point.
A backlog of 7,000 vehicles built up but the process of starting to clear those started on Tuesday.
Eurotunnel said anyone without a reservation or with a day trip ticket should not show up, as the priority was those who had been due to travel over the weekend.
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