British Airways has said it is aiming to operate a "near normal schedule" at Gatwick and the "majority of services" from Heathrow on Sunday after a global IT crash crippled the airline.
Thousands of passengers were left stranded after scores of planes were grounded by the major systems failure believed to have been caused by a power supply issue.
The airline said its engineers were "continuing to work hard" to restore its services and it hoped most UK flights would resume on Sunday.
However experts predict the disruption could continue for several days and BA is facing huge compensation costs after all its flights from Gatwick and Heathrow were cancelled on Saturday.
As IT teams tried to fix the system:
There were chaotic scenes at the Gatwick and Heathrow as people tried to make their way overseas for the long weekend and half-term school holiday.
All of BA's check-in and operational systems were affected by the issue, including the airline's customer service phone lines and rebooking function.
The incident had a knock-on effect on BA's operations around the world.
BA chief executive Alex Cruz said the airline was "extremely sorry" for the "huge inconvenience" suffered by customers, especially families heading on half term holidays.
Travellers have been told to check the airline's website and Twitter account for updates before setting off for the airport.
The glitch is believed to have been caused by a "power supply issue" and there is no evidence of a cyber attack, the airline said.
BA initially cancelled all flights before 6pm on Saturday but later announced that planes would be grounded for the rest of the day and warned passengers not to go to the airports.
It is feared that it could take days for services to return to normal and clear the backlog of passengers.
Air industry consultant John Strickland said the disruption could "run into several days" and added: "There's a massive knock-on effect.
"Customers and from the airline's point of view - manpower, dealing with the backlog of aircraft out of position, parking spaces for the aircraft - it's a challenge and a choreographic nightmare."
Several travellers at Heathrow told the Press Association they were not told their flights were cancelled until more than an hour after the airline put out a press statement announcing the decision.
Student Emily Wilson told the Press Association: "We were told (it would be) about three hours for collecting bags, that all compensation will have to be done online, and that we are unable to rebook flights now because of the system being down."
At Gatwick frustrated passengers could be seen surrounding BA staff at the check-in as they handed out letters which apologised for the cancellations and gave details about how to claim for hotels, local transport and refreshments.
Teacher Gemma Richardson, 30, who is 24 weeks pregnant, said "it was chaos" when she and her husband arrived at the airport with their two-year-old daughter.
"It seems that because it is a bank holiday weekend there is no spare flights. We are on standby but it is very unlikely we are going anywhere," she added.
Customers who saw their flights cancelled are being refunded or rebooked on to new services and other options are available for those who no longer want to fly.
A BA spokesman said: "We are continuing to work hard to restore all of our IT systems and are aiming to operating a near normal schedule at Gatwick and the majority of services from Heathrow on Sunday.
"We are extremely sorry for the huge disruption caused to customers throughout Saturday and understand how frustrating their experiences will have been.
"We are refunding or rebooking customers who suffered cancellations on to new services as quickly as possible and have also introduced more flexible rebooking policies for anyone due to travel on Sunday and Monday who no longer wishes to fly to/from Heathrow or Gatwick.
"We would advise customers travelling across the Bank Holiday weekend to continue checking the status of their flight on our website, www.ba.com before coming to the airport."
BA had issues with its online check-in systems in September and July last year, causing severe delays for passengers.
Page 2: 07:33
Passengers lucky enough to be aboard one of the few flights taking off on Saturday later found their hold luggage had not made it onto the plane with them.
Terry Page, 28, from London, flew from Heathrow to Fort Worth, Texas, where he and "about 50" others were told they would have to wait until Monday before being reunited with their bags, he told the Press Association.
A Heathrow spokesman said: "Because of the disruption yesterday, some passengers have had to leave Heathrow without their bags.
"We are working hard with BA to reunite passengers with their bags as soon as possible and are sorry for the inconvenience caused."
Passengers were advised to contact BA with any concerns regarding their luggage.
Departure boards showed flights departing from Gatwick early on Sunday, however several flights from Heathrow were cancelled.
A Heathrow spokesman said "delays and cancellations of British Airways flights are expected today", while the airline said there would be "some knock-on disruption to our schedules on Sunday, as aircraft and crews are out of position around the world".
Heathrow advised passengers whose flights were cancelled on Saturday not to travel to the airport unless they have already rebooked.