British Airways is aiming to operate a "near normal" schedule at Gatwick Airport and the majority of its flights from Heathrow tonight, after an IT failure caused chaos for its weekend flights.
But it said there would be some knock-on disruption to its schedules, as aircraft and crews were out of position around the world.
Serious problems with the airline's IT system led to thousands of passengers having their plans disrupted, after all flights from Heathrow and Gatwick were cancelled.
Passengers described "chaotic" scenes at the airports, with some criticising BA for a lack of information.
The airline apologised and said there was no evidence of a cyber attack.
BA chief executive Alex Cruz said: "We believe the root cause was a power supply issue."
In a video statement released via Twitter, he added: "I am really sorry we don't have better news as yet, but I can assure you our teams are working as hard as they can to resolve these issues."
The airline hoped to be able to operate some long haul inbound flights on Saturday, landing in London on Sunday, Mr Cruz added.
The GMB union has suggested the failure could have been avoided, had the airline not outsourced its IT work.
BA denied the claim, saying: "We would never compromise the integrity and security of our IT systems".
All passengers affected by the failure - which coincides with the first weekend of the half-term holiday for many in the UK - will be offered the option of rescheduling or a refund.
The airline, which had previously said flights would be cancelled until 6pm BST, has now cancelled all flights for Saturday and asked passengers not to come to Gatwick or Heathrow airports.
Other airlines flying in and out of the two airports are unaffected.
Architect and TV presenter George Clarke was stuck in Heathrow. He told the BBC it was one of the "most turbulent, badly organised days, that I've ever experienced in Britain".
"The lack of communication all day was woeful. There wasn't a single Tannoy announcement all day in the terminal, not a single member of staff came up to us," he said.
"The only time I found out my flight was cancelled was from the BBC News website."
'They can't do anything at all'
The problems affected BA call centres, the website and the mobile app.
Aviation expert Julian Bray said: "It's frozen the whole system so no British Airways plane can actually take off, they can't move the baggage, they can't issue passenger credentials, in fact they can't do anything at all.
"This is a very serious problem, they should have been able to switch to an alternative system - surely British Airways should be able to do this."
Malcolm Ginsberg, editor in chief at Business Travel News, expected the disruption to last for "three or four days".
BA aircraft landing at Heathrow are unable to park as outbound aircraft cannot vacate the gates, which has resulted in passengers being stuck on aircraft.
Mick Rix, GMB's national officer for aviation said: "This could have all been avoided.
"BA in 2016 made hundreds of dedicated and loyal IT staff redundant and outsourced the work to India... many viewed the company's actions as just plain greedy."
BA staff in Heathrow's Terminal 5 were resorting to using white boards, according to passenger Gareth Wharton.
Delays have been reported in Rome, Prague, Milan, Stockholm and Malaga due to the system failure.
Philip Bloom said he had been waiting on board a Heathrow-bound flight at Belfast for two hours.
He added: "We haven't been told very much just that there is a worldwide computer system failure.
"We were told that we couldn't even get on other flights because they are unable to see what flights we can be moved to."