An EgyptAir flight from Paris to Cairo made two sharp turns before plunging into the Mediterranean Sea, Greece's defence minister says.
Panos Kammenos said the Airbus A320 had "turned 90 degrees left and a 360-degree turn to the right" and dropped more than 6,700m (20,000ft) before disappearing from radar.
Sixty-six people were on board, most of them from Egypt or France.
A major search is under way in seas south of the Greek island of Karpathos.
Greek and Egyptian armed forces are involved in the effort, and France has offered to send boats and planes.
Of those on board, 56 were passengers, seven were crew members and three were security personnel. A Briton was among those on board.
Flight MS804 left Charles de Gaulle airport in Paris at 23:09 local time on Wednesday (21:09 GMT) and was scheduled to arrive in the Egyptian capital soon after 03:15 local time on Thursday.
EgyptAir said the plane had been flying at 11,300m (37,000ft) when it disappeared from radar shortly after entering Egyptian airspace.
Greek aviation officials say air traffic controllers spoke to the pilot when he entered Greek airspace and everything appeared normal.
They tried to contact him again at 02:27 Cairo time, as the plane was set to enter Egyptian airspace, but "despite repeated calls, the aircraft did not respond". Two minutes later it vanished from radar.
French President Francois Hollande said he was keeping an open mind about the cause: "We will draw conclusions when we have the truth about what happened.
"Whether it was an accident, or whether it was – and it's something that is on our minds – terrorism."
He has spoken to his Egyptian counterpart and both leaders have been holding emergency meetings with their top officials, according to reports from both countries.
Analysis by Frank Gardner, BBC security correspondent
The spectre of terrorism inevitably looms over this latest aviation tragedy.
While there is no evidence yet to indicate a malicious attack, it was only seven months ago that the Islamic State (IS) group planted a bomb that brought down a Russian passenger jet over the Sinai.
IS have vowed to continue targeting the Egyptian state and westerners who visit Egypt.
In a different but also malicious incident in 1999 a disturbed EgyptAir co-pilot put a Boeing passenger plane into a fatal dive off the US coast killing all 217 people onboard, though the Egyptian government continues to say it was a mechanical failure.
However at this stage investigators will be keeping an open mind about the cause of the plane's disappearance, including checking the maintenance record of the airframe.
Some of the relatives of those on board gathered at airports in Cairo and Paris to wait for news.
French Foreign Minister Jean-Marc Ayrault who met some of the relatives at a crisis centre in Paris Charles de Gaulle called it a "moment of intense emotion" for them.
Flightradar24 listed details of the plane's journey on Wednesday which showed it had flown from Asmara, in Eritrea, to Cairo, then on to Tunis, in Tunisia, before heading, via Cairo, to Paris.
Aviation analyst Alex Macheras told the BBC that Airbus A320s were regularly used for short-haul budget flights and had "an amazing safety record".
In March, an EgyptAir plane was hijacked and diverted to Cyprus. The attacker later surrendered and all hostages were released.
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