Passengers have been told not to travel to the London airports because of "extreme congestion" at the terminals, with all BA planes grounded before 6pm on Saturday.
A spokeswoman for the airline said: "We have experienced a major IT system failure that is causing very severe disruption to our flight operations worldwide."
BA has apologised to customers as scores of Britons headed overseas for the long weekend and half-term school holidays on Saturday morning.
A spokeswoman said: "We apologise to customers who are facing some delays following an IT outage this morning.
"We are working to resolve the problem as quickly as possible."
The airline has experienced problems with its online check-in systems in the past.
Passengers were hit by severe delays in September and July last year because of IT glitches.
The latest problem meant parts of BA's website were unavailable and some travellers claimed they could not check in on the mobile app, with many venting their frustration on social media.
One passenger, Henry Tail, tweeted: "Hey @British_Airways, couldn't log in to the app with my boarding pass on because systems were down and I missed my flight. What can I do?"
Another, Nadia Whittley, posted: "British airways computers down T5 and standstill, stranded on the plane and they make us pay for food!! Please retweet this indicency! (SIC)"
Julie Adie wrote: "We are on the runway.. For hour now..no offer of drinks..and because ba only takes cards now, we only have cash...wat we supposed to do!!?? (SIC)"
Melissa Davis, who runs MD Communications, a legal PR agency in London, was held for more than an hour and a half on the tarmac at Heathrow, on board a BA flight returning from Belfast.
Speaking from the plane, Miss Davis said the air conditioning had been off "so I don't think we will be going anywhere any time soon", but added that the passengers had been kept informed by their pilot and given water while they remained seated.
She later said she and others were then told they could not transfer to other flights because "they can't bring up our details".
Mr Tail, a 27-year-old teacher from London, claimed he had missed his flight to Rome because of the technical problems.
"I checked in online using the BA app at 8.15 for my flight at 9.25, then went and had breakfast," he told the Press Association.
"At some point the app restarted and when I went to go through security I couldn't log in to my booking to get my QR code.
"This meant I couldn't go through security, and by the time I'd gone back and forth to various customer service desks the flight had closed," he said.
"I've tried to call customer services but their systems are still down so they weren't able to help me, and they told me it was a worldwide issue."
Gareth Wharton, also at Heathrow Terminal 5, tweeted a picture of BA staff writing gate information on a whiteboard amid the systems outage.
"Gets worse, #T5 staff having to put gate info up on a white board #LowFi #Heathrow," he tweeted.
It comes a day after passengers at Gatwick Airport faced chaotic scenes and long queues due to a baggage system problem.
Those taking flights on Friday were forced to travel without their hold bags and were asked to carry any essential items in their hand luggage.