A gunman still on the run after killing three people near a Christmas market in Strasbourg has been identified as 29-year-old local man Cherif Chekatt.
French interior minister Christophe Castaner said there would be an increased police presence across France and that border security would be ramped up.
There are concerns the suspect may have escaped across the border into Germany.
As well as the three deaths, 13 people were injured, eight seriously, when gunshots were heard just after 8pm near the market in Place Kleber, which attracts millions of tourists each year.
Sky sources have confirmed that one of the three victims killed was a Thai tourist.
Witnesses earlier said he was shot in the head and did not respond to attempts to revive him.
The Thai embassy in Paris named him as 45-year-old Anupong Suebsamarn. It said he had just arrived in Strasbourg and was planning to travel to Paris on Thursday.
Police with machine guns were seen running into the square moments after the first shots were fired and it was quickly evacuated.
Officers chased the gunman for more than two hours before they cornered him and there was a shootout.
The suspect was wounded, along with a soldier, but Chekatt managed to get away.
Five people have so far been detained as part of the investigation.
The 29-year-old was born in Strasbourg and police said he is a suspected robber who was due to be arrested on Tuesday morning.
His home was searched and they found explosive materials, but he was not there.
Authorities said Chekatt was an "S file" security risk - a list of anyone in France suspected of being a radical.
There was a police operation near the city's cathedral on Wednesday morning but witnesses reported it quickly ended and officers were seen leaving. No more details were immediately available.
A terror investigation has been opened for murder and attempted murder, the French prosecutor's office said.
A Sky News source said MEPs were among many people stuck in buildings in Strasbourg in the hours after the shootings because of a security lock down.
Emmanuel Foulon, a press officer for the European Parliament - just two miles away, was in the square and said there was panic as everyone took to the ground for cover.
Sinn Fein MEP Martina Anderson, who had just left the market, told Sky News: "We were walking towards the scene of where the shooting took place.
"I heard over six shots and at that stage, there was chaos, confusion, everyone was running - there was pandemonium.
"There were some people with children walking towards the incident and we were telling them to move back.
"Most people were shocked, confused - there was pandemonium - people didn't understand what was happening - we hadn't [either]. We had heard the shots quite clearly and there were people running away from the scene."
French president Emmanuel Macron is sending his interior minister, Christophe Castaner, to the eastern French city, an Elysee palace official said.
Theresa May said she was "shocked and saddened" by the attack and that her thoughts are "with all those affected and with the French people".
Jean-Claude Juncker, president of the European Commission, said his thoughts are with the victims and that Strasbourg is a symbol of the peace and democracy of Europe.
"Values that we will always defend. The Commission stands alongside France," he said.
Sky News' Europe correspondent Mark Stone said that an airport-style security system to get on to the island where the market is held - in place two years ago - is no longer there, but police and soldiers were at the scene.
Strasbourg Christmas market is one of the oldest in Europe, with 300 wooden stalls set up in the city's historic centre from 23 November to Christmas Eve.
Ten suspected Islamic militants plotted to blow up the market on the millennium and were jailed for up to nine years in 2004.
France remains on high alert following a series of Islamic State inspired attacks in 2015 and 2016, which killed more than 200 people.
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