British Airways has cancelled all flights from Heathrow and Gatwick until 18:00 BST because of computer problems.
A "major IT system failure that is causing very severe disruption to our flight operations worldwide", the airline said.
It apologised for the global system outage and said it was working to resolve the the problem.
Heathrow Airport said it was "working closely" with BA to solve the issue.
There is no evidence at this stage to suggest the system failure was caused by a cyber attack, British Airways says.
All affected passengers will be offered the option of rescheduling or a refund.
In a statement, the airline asked passengers with flights before 18:00 BST on Saturday not to come to Gatwick or Heathrow airports, which it said had become extremely congested.
Other airlines flying in and out of Heathrow and Gatwick are unaffected.
The problems mean parts of British Airways's website are unavailable and some travellers claimed they could not check in on the mobile app.
BA aircraft landing at Heathrow are unable to park up as outbound aircraft cannot vacate the gates, which has resulted in passengers being stuck on aircraft.
A passenger said he had been sitting on a plane at Heathrow for 90 minutes. He said the captain told passengers the IT problems were very problematic and catastrophic.
BA staff in Heathrow's Terminal 5 were resorting to using white boards, according to passengers.
Delays have been reported in Rome, Prague, Milan, Stockholm and Malaga due to the system failure.
Passengers say they haven't been told very much just that there is a worldwide computer system failure.
They were told that we couldn't even get on other flights because they are unable to see what flights we can be moved to.
Not even flights from other countries can fly to the two airports.
British Airways staff are unable to say how long delays would take, but they say all flights are grounded around the world.
Passengers are being told their computer systems were down worldwide.
Some delayed passengers have been given vouchers for food and drink.
EU flight delay rights
- If your flight departed the European Union or was with a European airline, you might have rights under EU law to claim if the delay or cancellation was within the airline's control
- Short-haul flights: 250 euros for delays of more than three hours
- Medium-haul flights: 400 euros for delays of more than three hours
- Long-haul flights: 300 euros for delays of between three and four hours; and 600 euros for delays of more than four hours
- If your flight's delayed for two or more hours the airline must offer food and drink, access to phone calls and emails, and accommodation if you're delayed overnight - including transfers between the airport and the hotel.
I hope this is just not a cyber attack or