British Airways says it's aiming to operate a "near normal schedule" at Gatwick and the "majority of services" from Heathrow on Sunday (local time) after a global IT crash crippled the airline.
Thousands of passengers were stranded on Saturday after scores of planes were grounded by the major systems failure believed to have been caused by a power supply issue.
The airline said it was "extremely sorry" for the "huge disruption" it cause and that its engineers were continuing to work hard to restore its services.
The failure, caused by a power supply problem, disrupted BA's flight operations worldwide and also hit its call centres and website, said Alex Cruz, the chair and chief executive of BA, part of Europe's largest airline group IAG.
"All of our check-in and operational systems have been affected and we have cancelled all flights from Heathrow and Gatwick for today," Mr Cruz said in a video message on Twitter on Saturday (local time).
"We are extremely sorry for the huge inconvenience this is causing our customers and we understand how frustrating this must be, especially for families hoping to get away on holiday."
He said the airline's IT teams were working "tirelessly" to fix the problem and there was no evidence of any cyber attack.
Several hours after problems began cropping up on Saturday morning, BA suspended flights up to 6pm because the two airports had become severely congested. The airline later scrapped flights from Heathrow and Gatwick for the rest of the day.
The problems, which passengers said had affected flights across Britain, came on a particularly busy weekend with a public holiday on Monday and many children starting their school half-term breaks.
Terminals at Heathrow and Gatwick became jammed with angry passengers, with confused BA staff unable to help as they had no access to their computers.
BA is the latest airline to be hit by computer problems. Last month Germany's Lufthansa and Air France suffered a global system outage which prevented them from boarding passengers.
In September last year BA apologised to passengers for check-in delays caused by operational glitches that delayed flights at Gatwick and Heathrow, in a repeat of a similar incident that affected London-area flights for the airline last July.
BA said it would try to get affected customers onto the next available flight although the re-booking process was being hindered by the system problems. Those unable to fly would get a full refund, Mr Cruz said.
"We hope to be able to operate some long-haul inbound flights tonight, which will land in London tomorrow," he said.
Some passengers said they had boarded flights but were then left stuck on the runway.
One traveller, PR executive Melissa Davis, said she was held for more than an hour and a half on the tarmac at Heathrow aboard a BA flight arriving from Belfast.
She said passengers had been told they could not transfer to other flights because "they can't bring up our details."
Passenger Phillip Norton tweeted video of an announcement from a pilot to passengers at Rome's Fiumicino airport, saying the problem affects the system that regulates what passengers and baggage go on which aircraft.
Airlines depend on huge, overlapping and complex IT systems to do just about everything, from operating flights to handling ticketing, boarding, websites and mobile-phone apps. Some critics say complex airline technology systems have not always kept up with the times.