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.
BA faces paying out millions of pounds in compensation after the failure of its computer systems led to long delays and cancellations over the weekend.
Passengers are facing a third day of disruption as the airline deals with the impact of a worldwide IT crash.
There are EU regulations governing compensation for cancelled flights leaving from EU airports.
The amount of money reimbursed depends on the length of delay and whether it is a short, medium or long-haul flight.
A BA spokesman said: We have been giving letters to customers telling them how to claim under EU compensation rules and we will fully honour our obligations.
Compensation could be more than £500, depending on the distance of the flight.
But compensation is not automatic. customers have to write a complaint letter to the airline.
Some airlines, and consumer body Which? provide standard compensation application letters for passengers to complete.
Passengers are entitled to assistance and compensation, if the disruption was within an airline's control
This applies to short-haul flights delayed by at least two hours,and to medium-haul flights delayed by three hours, or long-haul delayed by four
For overnight delays, airlines must provide hotels, and transfers between airport and hotel.
Airlines have to offer full refunds, paid within seven days, or re-bookings for a flight cancelled at short notice
In addition passengers can also claim compensation.
Cancellation amounts are: 250 euros (£218) for short-haul, 440 euros (£384) for medium-haul and 600 euros (£523) for long-haul.
The EU regulations do not apply if the disruption has been caused by factors outside the airline's control, such as a strike.
In 2014, two UK Supreme Court judgements stated airlines should have to pay out when a delay was caused by a technical fault, which appears to have happened in this instance with BA.
Aviation expert Julian Bray adds:Also remember if you actually paid for your fare with a credit card, then the Consumer Credit Act comes into play, and you could well get money back that way.
Meanwhile, BA says it is meeting its obligations in providing hotel accommodation and refreshments for customers whose journeys have been disrupted.
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 who was due to travel on Saturday, Sunday, today [Monday] and Tuesday who no longer wishes to fly to-from Heathrow or Gatwick,they add.
Customers on flights that have been cancelled can claim a full refund or rebook to a future date for travel up until the end of November 2017.
Customers should get in touch with us directly via our Manage My Booking tool on ba.com or our contact centres so that we can re-book or re-route them to their destination as quickly as possible.
They added: Passengers whose journeys are disrupted have been urged to keep any food, transport or accommodation receipts and can make a claim in due course through our customer relations teams.
The firm says it has been booking passengers onto other airlines where necessary.
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.
Laura Thomson and Sam Sciortino, from Woking, planned a Sunday ceremony in Santorini but will now marry on Monday.
The bride-to-be's brother and family - including three bridesmaids - waited at Heathrow for 13 hours on Saturday before leaving without their luggage.
They have now booked flights with a different airline.
Ms Thomson said: "We have had to contact suppliers, and change all the arrangements we had planned.
"We've been planning our big day for two years only to be forced to rush around and change everything to take place a day later.
The couple flew to Santorini earlier in the week, ahead of their guests. They said an aunt and uncle were also caught up in the BA disruption.
Ms Thomson said: My brother is one of the groomsmen, and my nieces and sister-in-law are my bridesmaids, so naturally I wanted them to share the day with us.
They booked with British Airways because they thought it was a reliable company.
Ms Thomson's sister-in-law Marcia Thomson said:We are devastated that this has happened.
Our luggage had already been checked in and after waiting for 13 hours at the airport we were forced to leave without it.
Luckily we have managed to get flights to Crete this evening with a different airline.
We will have to get a ferry from there to Santorini on Monday morning just in time for the wedding, but without our bridesmaids dresses.
Posting on Facebook Ms Thomson shared her disappointment with family and friends.
I am just a shell of myself right now. We are absolutely shattered, she said.
I have been in tears with our dream of how it should have been slowly fading away, knowing I can't share the day with my family the way I imagined.
She added:The hotel have been very accommodating but we've incurred extra costs as all the food has had to be paid for again.
We had a family cruise booked which also had to be cancelled and we have lost money on that as well.
Mr Sciortino said: It's one of those things you don't think will happen to you but we've been so lucky that we have managed to rearrange it even though there are certain things that we will be missing.
The British Airways check-in hall at Gatwick Airport is eerily empty. Elsewhere people are milling around with luggage, gazing hopefully at information boards not knowing if they will be going on holiday.
The emotion came earlier with the first announcement that a major IT system failure meant flights had been cancelled, both here at Gatwick and at Heathrow until 18:00 BST. BA would later announce no flights would be leaving from either airport for the rest of Saturday.
People were in tears as staff started to clear a crowded check-in queue.
Some holidaymakers have been here all day, others have just turned up hoping to catch a flight. BA staff are on hand to help explain the situation as some passengers try to rethink their plans.
Among the usual security warnings to stay with your bags there have been announcements apologising for the disruption. Many people are waiting calmly and patiently to see what happens.
Fiona Lancaster, 49, from Braintree, Essex, was due to fly to Sicily and has been in the airport for four hours.
They don't automatically rebook you. You have to try to rebook yourself and if you book with another airline there is no refund.
I've been deciding whether to sleep in the car, she said.
There have been a lot of people in tears. I'm trying not to cry. I was going on holiday for a week and am a bit upset. I was supposed to be on Mount Etna for my 50th birthday next Friday, I was on a tour.
Mariah McIntyre, 39, who is with her two children Sloane, three, and Ellie, seven, were supposed to be flying to Portugal for a half-term holiday.
Ms McIntyre said: We knew the situation but we still turned up. It was wishful thinking. We are going to check other flights.
We'll give it several hours and will await the next update, we are looking at other airlines and we're trying to be resourceful and we are going to look at other routes via Lisbon and Seville.
I had a feeling about the refunds. We're kind of out of luck but we have to be positive and we have to try and not get frustrated until we've tried every which way.
Paul said: "We arrived at 07:45. We are going to Orlando for a once-in-a-lifetime trip to Disney.
We were waiting to board the plane at the gate when we got told. We got turned back to departures and told to book a flight and stay in a hotel and pay for everything again.
It didn't help us in any way. They're not putting us in a hotel. They didn't tell us what was happening, we just found ourselves in the baggage hall.
We saved for a year for this. it's a £15,000 holiday and we have probably lost three days of it.
Builder Kujdes, 31, said: The website's crashed. I just got here I don't know much about what's happening. I'm trying to fly if I can. But they said there is no information. I talked to one of the managers.
He said that the system had crashed I had to wait until 18:00 and and there is no more information. I'm flying to Albania for a holiday. I just got here I don't know what's happening I don't even know whether I've got to rebook.
Dr Tikare had been due to fly to Malaga with his family and said they were feeling pretty anxious.
We're coming back on Thursday, so it's not a long trip. I couldn't get any information on the BA site, so I've been checking for possible updates.
He said passengers had been speculating over whether the problems had stemmed from a cyber attack although BA has said there is no evidence that is what caused it.
Dr Tikare, who works as a paediatric doctor, said he had heard about people being stuck on planes for hours.
We are just hoping it will have cleared up and feeling optimistic, he added.
BA says that: The root cause was a power supply issue which our affected our IT systems - we continue to investigate this.
The airline said it could not add anything further at this stage, but it is understood that all systems are not fully up and running yet.
John, air transport expert and director at JLS Consulting, says: The problem has affected multiple parts of the business which are not only customer-facing, but also operational-facing, and without which the airline could not do many tasks, for example completing load sheets,which are needed for fuel calculationsfor aircraft."
The GMB union has suggested the BA computer systems failure was another example of the shortcomings of BA IT systems since they made a number of staff redundant, and outsourced their work to India in 2016.
Mick Rix, GMB national officer for aviation said: BA made hundreds of dedicated and loyal IT staff redundant and outsourced the work to India.
BA have made substantial profits in for a number of years, and many viewed the company's actions as just plain greedy.
Meanwhile aviation expert Mr Strickland, added: Surely a business should be able to make an outsourcing decision without any problems, if it is done in a quality-controlled way.
But this issue is part of the analysis that will have to be done by BA.
But the airline says: We would never compromise the integrity and security of our IT systems. IT services are now provided globally by a range of suppliers and this is very common practice across all industries and the UK government.
British Airways employs around 35,000 people in the UK providing high skilled and well paid jobs. It hires 1,000 people a year and has a strong apprenticeship programme.
BA says: Customers on flights that have been cancelled can claim a full refund or rebook to a future date for travel up until the end of November 2017. Customers are urged to keep any food, transport or accommodation receipts and can make a claim in due course through our Customer Relations teams.
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 have provided customers with hotel accommodation.
The best channel for customers to use to get information about their flight is Manage My Booking on ba.com. This is now updated regularly.
We have also been using social media to communicate, and airport communication channels. Our CEO video which was posted on Twitter, YouTube and Faceboo has received more than 175,000 views.
We are extremely sorry for the disruption caused to customers and understand how much frustration this is causing.
Delayed travellers will also be able to claim financial compensation under EU law, unless the disruption has been caused by factors outside the airline's control.
In 2014 two UK Supreme Court judgements stated airlines should have to pay out when a delay was caused by a technical fault.
BA passengers trying not to cry.
BA says: We have been doing everything possible to provide as much information to customers as possible, but it has been challenging.
Many of the systems our staff usually use have not been functioning properly which has meant we were slower than usual to give customers accurate information at the airports.
In addition, unfortunately the systems that we use to send emails and texts to individual customers about their flights have also been affected by the IT problems, so we haven't been able to communicate with customers in our usual ways.
There had been complaints from passengers on Saturday that they had not been informed their flights were cancelled until after the airline had put out a media statement announcing the decision.
Passenger Terry, 28, from London said: There's no such announcement here. The boards are showing go to gate, and no mention of cancellations.
Communications complaints continued at Heathrow on Sunday, with one passenger saying they had no idea how much longer we'll be here and we're getting no communication from the staff.
Aviation expert John Strickland was at Heathrow on Saturday to collect a friend flying in from Mexico, and says : Communications have been a challenge.
There were BA people there on Saturday doing what they could, but the tools usually at the disposal of staff were not available.
We are becoming more and more reliant on automation, even in things like communications, and less on actual people. It means it is a challenge for businesses when things go wrong.
Could it happen again? What has BA learned ?
John says: They will have to sit down in the cold light of day and analyse the causes and how to eliminate them in future.
There will need to be in-depth planning in terms of managing any similar potential situation again.
BA is used to things like bad weather, or air traffic control issues, and other challenges, but this is different. By its very nature they did not have the information available that they would have liked to have had.
But he also added: This could have happened to any other airline, including a low-cost one.