British Airways is working to restore its computer systems after a power failure caused major disruption for thousands of passengers worldwide.
The airline is "closer to full operational capacity" after an IT power cut resulted in mass flight cancellations at Heathrow and Gatwick.
Thousands of passengers remain displaced, with large numbers sleeping overnight in terminals.
BA has not explained the cause of the power problem.
So far on Monday, 13 short-haul flights at Heathrow have been cancelled.
Heathrow advised affected BA passengers not to travel to the airport unless their flights had been rebooked, or were scheduled to take off today.
Passengers on cancelled flights have been told to use the BA website to rebook.
Chief executive Alex Cruz has posted videos on Twitter apologising for what he called a "horrible time for passengers".
But no-one from the airline has been made available to answer questions about the system crash, and it has not explained why there was no back-up system in place.
Cancellations and delays affected thousands of passengers at both Heathrow and Gatwick on Saturday.
All flights operated from Gatwick on Sunday but more than a third of services from Heathrow - mostly to short-haul destinations - were cancelled.
Passengers slept on yoga mats handed out by the airline as conference rooms were opened to provide somewhere more comfortable to rest.
What went wrong at BA?
Rory Cellan-Jones, technology correspondent
BA blames a power cut, but a corporate IT expert said it should not have caused "even a flicker of the lights" in the data-centre.
Even if the power could not be restored, the airline's Disaster Recovery Plan should have whirred into action. But that will have depended in part on veteran staff with knowledge of the complex patchwork of systems built up over the years.
Many of those people may have left when much of the IT operation was outsourced to India.
One theory of the IT expert, who does not wish to be named, is that when the power came back on the systems were unusable because the data was unsynchronised.
In other words the airline was suddenly faced with a mass of conflicting records of passengers, aircraft and baggage movements - all the complex logistics of modern air travel.
BA said it operated virtually all scheduled long-haul flights on Sunday, but the knock-on effects of Saturday's disruption resulted in a reduced short-haul programme.
"We apologise again to customers for the frustration and inconvenience they are experiencing and thank them for their continued patience."
Ian Sanderson, one of the affected passengers who is stuck in transit in London, said he was "incandescent with rage" after being unable to rebook his flight, or speak to a member of staff.
Speaking on Sunday evening, he said: "I've bombarded them with about 100 tweets in the last 24 hours. I know that's annoying but there's nothing else I can do.
"We've tried to call them on the numbers they give and all we've got is the same recorded message which then cuts off at the end."
Former Virgin Airlines spokesman Paul Charles said: "What seems remarkable is there was no back-up system kicking in within a few minutes system failing.
"Businesses of this type need systems backing up all the time, and this is what passengers expect."
BA is liable to reimburse thousands of passengers for refreshments and hotel expenses, and travel industry commentators have suggested the cost to the company - part of Europe's largest airline group IAG - could run into tens of millions of pounds.
Shares in IAG listed on the Madrid stock exchange are currently trading down by about 3%.
Customers displaced by flight cancellations can claim up to £200 a day for a room (based on two people sharing), £50 for transport between the hotel and airport, and £25 a day per adult for meals and refreshments.
Consumer expert Franky Brehany said travellers stranded in a "high-value city" like London may be able to claim more and should keep all receipts.
But he added that it might be harder for passengers to claim compensation, as BA may blame "extraordinary circumstances" - "like an act of God or force majeure" - meaning the airline would only have to reimburse hotel and food costs.
Thousands of bags remain at Heathrow Airport, but BA has advised passengers not to return to collect them, saying they will be couriered to customers.
The airline said there was no evidence the computer failure was the result of a cyber-attack. It denied claims by the GMB union that the problem could be linked to the company outsourcing its IT work.
Gatwick Airport said it was continuing to advise customers travelling with British Airways to check the status of their flight with the airline before travelling to the airport.
EU flight delay rights
- If your flight departed from within 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.