The airline has apologized for what it calls an "IT systems outage" and says it is working to resolve the problem.
BA staff in Heathrow's Terminal 5 were resorting to using white boards, it added.
London's Heathrow Airport, one of the world's busiest, said it was working with BA and advised passengers to check their flight status before travelling.
British Airways says it is canceling all of its flights from London's Heathrow and Gatwick airports for the rest of the day because of a "major IT system failure".
BA aircraft landing at Heathrow are unable to park up as outbound aircraft can not vacate the gates, which has resulted in passengers being stuck on aircraft.
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.
All passengers affected by the failure - which coincides with the first weekend of the half-term holiday for many in the United Kingdom - will be offered the option of rescheduling or a refund.
American Airlines, which operates code-share flights with BA, said it was unaffected.
One passenger tweeted: "Stuck on a British Airways plane at Belfast, going nowhere".
Air industry consultant John Strickland said the disruption could "run into several days" and added: "There's a massive knock-on effect".
"Customers and from the airline's point of view - manpower, dealing with the backlog of aircraft out of position, parking spaces for the aircraft - it's a challenge and a choreographic nightmare". "No updates or staff around", travel writer and blogger @WildWayRound tweeted.
Earlier, passengers at Heathrow reported long lines at check-in counters and flight delays.
In August a power surge near USA airline Delta's Atlanta headquarters caused computers to crash and led to widespread delays across Delta's entire network.
After the recent outages, outside experts have questioned whether airlines have enough redundancy in their huge, complex IT systems and test them frequently enough.