The problems, which passengers said were affecting flights across the country, came on a particularly busy weekend in Britain with a public holiday on Monday and many children starting their school half-term breaks.
"We apologise to customers who are facing some delays following an IT outage this morning," a British Airways spokeswoman said. "We are working to resolve the problem as quickly as possible."
She gave no further details. Passengers on social media reported long delays at check-in desks and flights being held on runways.
"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'."
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 BA staff basically saying 'if you've not checked in online, you've missed your flight'," she told Sky News.
British Airways is the latest airline to be hit by computer problems. Last month Lufthansa and Air France suffered a global system outage which prevented them from boarding passengers.
(Reporting by Michael Holden; Editing by Toby Chopra and Andrew Bolton)