Serious problems with British Airways' IT systems have 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 has apologised, and told passengers not to come to the airport.
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."
Mr Cruz said there was no evidence the computer problems were the result of a cyber attack.
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 18:00 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."
The problems have 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, expects 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.
Journalist Martyn Kent said he had been sitting on a plane at Heathrow for 90 minutes. He said the captain told passengers the IT problems were "catastrophic".
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."
Analysis - By Richard Westcott, BBC transport correspondent
As ever, it is a lack of information that is really making BA passengers angry… we're still awaiting an explanation from the airline and a timescale for how long the problems might last.
The GMB union says this meltdown could have been avoided if BA had not made hundreds of IT staff redundant and outsourced their jobs to India at the end of last year.
Yes, the union has a big axe to grind, but people will want to know if the airline made its IT systems more vulnerable by scaling back computer support to save money - although BA has just flatly denied it to me.
IT problems ripple through an airline. If planes cannot take off, they cannot leave gaps at the gate for others to land.
If flights are delayed by more than around five hours, the airline must swap crews because shift lengths are strictly limited for safety reasons.
Telling customers to stay away is a drastic measure, but it is the only chance BA has of clearing the backlog of flights.
The BBC's Phillip Norton was at Rome international airport, waiting to fly to London.
He said BA staff were unable to say how long delays would be, telling him "all flights are grounded around the world".
Alma Saffari was in Marseille waiting to get her flight back to Heathrow.
She said: "When we finally boarded the captain came out and told us their computer systems were down worldwide.
"Eventually after sitting on the tarmac for one and a half hours we disembarked the plane.
"Now we are sitting in the departure area outside the gate."
Ms Saffari, who is with her 13-month-old baby, said she had been given a voucher for food and drink.
EU flight delay rights
- If your flight departed the European Union or was with a European airline, you might have rights under EU law to claim if the delay or cancellation was within the airline's control
- Short-haul flights: 250 euros for delays of more than three hours
- Medium-haul flights: 400 euros for delays of more than three hours
- Long-haul flights: 300 euros for delays of between three and four hours; and 600 euros for delays of more than four hours
- If your flight's delayed for two or more hours the airline must offer food and drink, access to phone calls and emails, and accommodation if you're delayed overnight - including transfers between the airport and the hotel