EgyptAir Flight MS804 disappeared from radar 10 miles inside Egyptian air space at 2.30am Cairo time (1.30am BST) after taking off just under three-and-a-half hours earlier from Charles de Gaulle Airport.
Reports claim it crashed near Karpathos, around 50 miles east of Crete in the eastern Mediterranean.
French President Francois Hollande confirmed the aircraft had come down, saying: “I have been informed that the aircraft that left Paris to go to Cairo has been lost. It crashed.”
He added that “no hypothesis is excluded” in relation to the cause. Mr Hollande had earlier spoken with Egyptian President Abdel-Fattah el-Sissi by telephone, and they agreed to “closely co-operate to establish the circumstances”.
The UK Foreign Office said it was working with Egyptian and French authorities, with a spokesman adding: “Our staff are in contact with the family of a British national believed to be on board and are providing support.”
A major search and rescue operation has been launched by Egyptian and Greek authorities to find the remains of the aircraft, with reports of a flash in the sky over the Mediterranean.
Among those on board were a child and two babies, EgyptAir said.
The airline added that the country’s prime minister Sherif Ismail had arrived at its Cairo Airport crisis centre and received a detailed briefing.
The airline said the 56 passengers included 30 Egyptians, 15 French, two Iraqis and one each from Britain, Sudan, Chad, Portugal, Algeria, Canada, Belgium, Kuwait and Saudi Arabia.
Flight MS804 left Charles de Gaulle Airport at 11.09pm local time (10.09pm BST).
French foreign minister Jean-Marc Ayrault said France had offered military assistance to Egypt to help with the search.
There was confusion over whether a distress signal had been sent by the Airbus A320.
Egypt’s civil aviation authority said one was received at 4.26am local time, believed to be an automated message rather than one sent by the pilot.
However the Egyptian military later said it had received no distress message from the aircraft, in a statement on its website.
In March, a domestic EgyptAir flight with 72 passengers on board had to make an emergency diversion to Larnaca, Cyprus, after an alleged hijacking.
In October last year, 224 people were killed when a Russian aircraft crashed over Egypt’s Sinai Peninsula minutes after it took off from the Red Sea resort of Sharm el-Sheikh.
EgyptAir has provided free contact numbers for families concerned for relatives. From outside Egypt, anyone concerned should call + 202 2598 9320.
Greek authorities have deployed two aircraft, a C-130 military plane and one early warning aircraft, the Hellenic National Defence General Staff said.
It also said a frigate was heading to the area and helicopters were on standby on Karpathos for potential rescue or recovery operations.
The Airbus A320 was built in 2003 and was flying at 37,000ft, the airline said on Twitter.
It tweeted that the pilot had logged 6,275 flying hours, including 2,101 hours on the A320, and the co-pilot had logged 2,766 hours.