The airline apologised to customers for the issue, which is thought to have been caused by a problem with the IT system's power supply.
In a statement released on Sunday, chief executive Alex Cruz said: "I know this has been a horrible time for customers. We're not there yet, but we are doing our very best to sort things out for you."
The BBC's Phillip Norton, who has been stranded at Rome airport since Saturday, has been told he won't be able to fly back to London until Tuesday.

The airline is liable to reimburse thousands of passengers for refreshments and hotel costs.
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.
One traveller from Seattle said she had spent the evening sleeping on the floor of a hotel conference room.
Ashley Tracey, who was trying to get to Mumbai for her friend's wedding, said she had been queuing to rebook her flight for six hours.
She said: "There's no information I can't seem to get through online, I don't live here so I don't have a phone that works here."
line break
At the scene

On Sunday morning, Heathrow Terminal 5 descended into chaos again as people arrived to rebook flights after theirs were cancelled yesterday as well as new arrivals hoping they had not been affected by the company's IT failure.
Many people told me about the lack of knowledge or information provided by BA staff.
One American woman on a stop-over described how an employee asked what the queue they were standing in was for.
Some passengers slept on yoga mats provided by the airline as conference rooms were opened to provide somewhere more comfortable to rest.
A number of people arrived at the airport for 05:00 and were still none the wiser by lunch time whether they'd be home any time soon.
As the day has progressed, British Airways appears to have more of a grip on the situation. The departure lounge has calmed down and many have rearranged their flights.
But the terminal is still full of hour-long queues and for many this wasn't how they had hoped to spend their Bank Holiday weekend.
line break
There have been reports that some passengers who departed from Heathrow on Saturday found their luggage was not at their destination when they landed.
Terry Page, 28, arrived in Fort Worth, Texas after delays and said "about 50" passengers did not have their check-in luggage.