British Airways said there was no evidence the problem had been caused by a cyber attack.
The airline said terminals at Heathrow and Gatwick had become extremely congested because of the IT failure and all British Airways flights scheduled before 1700 GMT had been cancelled.
"Please do not come to the airports. We have experienced a major IT system failure that is causing very severe disruption to our flight operations worldwide," the airline said in a statement.
"We are extremely sorry for the inconvenience this is causing our customers and we are working to resolve the situation as quickly as possible."
The problems, which passengers said were 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.
British Airways 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.
"Still on the tarmac at Leeds. #britishairways reckon Heathrow is so backed up we can't set off. No way we'll make our Vegas flight," one passenger David Raine wrote on Twitter.
Another, journalist Martyn Kent, wrote: "Sat on plane at Heathrow for hour and a half now. @British_Airways Captain describes IT problem as 'catastrophic'."
London's Heathrow Airport, one of the world's busiest, said it was working with British Airways. Passenger Roshni Burt, who was flying from Heathrow to Bahrain with her young son, said there was no news about when her flight would depart.
"When we left the check-in area there were angry people, people getting frustrated that their flights were coming up or near to departure, people getting turned away ... with British Airways staff basically saying 'if you've not checked in online, you've missed your flight'," she told Sky News.
(Editing by Andrew Bolton and David Clarke)