BA said there was no evidence the problem had been caused by a cyber attack.
"The terminals at Heathrow and Gatwick have become extremely congested and we have cancelled all flights from Heathrow and Gatwick". As well as chaos at airports, phonelines are also jammed as travellers try to find out whether their flights have been affected.
Experts predict the disruption could continue for several days and the airline is facing huge compensation costs after all BA flights from Gatwick and Heathrow were cancelled on Saturday. "No way we'll make our Vegas flight", one passenger David Raine wrote on Twitter. "We've tried all of the self check-in machines; none were working, apart from one".
Melissa Davis, who runs a legal PR agency in London, was held for more than an hour and a half on the tarmac at Heathrow, on a BA flight returning from Belfast.
Heathrow officials said they were working with British Airways, which is based in the airport's Terminal 5, "following an issue with their IT system this morning that has caused some delays for passengers".
The airline did not say what caused the computer problem, only that it was working to resolve it as quickly as possible. He said passengers on planes that had landed at Heathrow were unable to get off because there was nowhere to park.
Heathrow said the IT problem had caused "some delays for passengers" and it was working with BA to resolve it.
If you are flying with BA this weekend, or have any flights to or from Heathrow or Gatwick, check with the airport first to ensure that your flight has not been cancelled or severely delayed.
The move to scrap all flights came at the start of a bank holiday weekend and during half-term school holidays.
Most long-haul flights due to land in London tomorrow (Sunday, May 28) are expected to arrive as normal, according to the spokeswoman.
This is the sixth major problem technical problem to hit British Airways this year.
"Hey @British-Airways, couldn't log in to the app with my boarding pass on because systems were down and I missed my flight".
"Manpower, dealing with the backlog of aircraft out of position, parking spaces for the aircraft -it's a challenge and a choreographic nightmare", he said.
In August 2016, Delta planes around the world were grounded when an electrical component failed and led to a shutdown of the transformer that provides power to the airline's data center.