The teacher who was beheaded in a street in France had received threats after showing controversial cartoons of the Prophet Muhammad to his pupils, French media report.
He has been named as 47-year-old Samuel Paty, a history and geography teacher.
Nine people have been arrested, including the parents of a child at Mr Paty's school, judicial sources are quoted as saying.
Police say the attacker was an 18-year-old man of Chechen origin.Â
The killing took place while a trial is under way in Paris over a 2015 Islamist assault on the French satirical magazine Charlie Hebdo, which was targeted for publishing the cartoons.
President Emmanuel Macron said the attack bore all the hallmarks of an "Islamist terrorist attack" and the teacher had been murdered because he "taught freedom of expression".
Speaking at the scene hours after the incident, he stressed national unity. "They will not prevail, they will not divide us," he said.
What do we know?
The attack occurred at about 17:00 (15:00 GMT) near the College du Bois d'Aulne, where he taught, in the town of Conflans-Sainte-Honorine, some 30km (20 miles) north-west of central Paris.
A man wielding a large knife attacked the teacher in the street, cutting off his head. Witnesses are said to have heard the attacker shout "Allahu Akbar", or "God is Greatest".Â
A picture of the beheaded teacher was posted to social media in the immediate aftermath, but it is not clear if this was by the assailant or an accomplice.Â
The attacker then ran from the scene but was chased by local police who had been alerted by the public.
The officers confronted the man in the nearby town of Éragny.
When they shouted at him to give himself up, he is said to have threatened them. The officers shot him and he died a short time later.Â
Police have said the attacker was an 18-year-old, born in Moscow but from Russia's predominantly-Muslim southern region of Chechnya. He was living in Évreux in Normandy. His grandparents and brothers are among those arrested.
Teachers across France told journalists of their fear, stunned by the targeting of a colleague murdered in broad daylight in a quiet suburban street.
Aside from the horror of this killing, there's powerful symbolism too. The education minister said the root of what happened was "hatred of the Republic".
The passing on of French national values - liberty, equality, fraternity - is seen as a core task of the education system here.Â
Three weeks after an attack on the former offices of Charlie Hebdo, this latest killing is yet more proof of the fault-lines over secularism and tolerance, which have left blood before in the country's streets.
Who was the victim?
Samuel Paty, a father, had taken a class about freedom of expression earlier in the month, reports say.
He is said to have shown the cartoons of the Prophet Muhammad while talking about the Charlie Hebdo case, and had reportedly advised Muslim students to leave the room if they thought they might be offended.
Some Muslim parents complained to the school and at least one reportedly took to social media to call for Mr Paty's sacking. French media report that Mr Paty had received a number of unspecified threats in the days after taking the class.Â
Flowers were being laid at the entrance to the College du Bois d'Aulne on Saturday.
Students were said to be distraught at the brutal killing of a well-liked teacher. One father wrote on Twitter that his daughter "is in pieces, terrorised by the violence of such an act. How will I explain to her the unthinkable?"Â
One of Mr Paty's former students, Martial, 16, said he had loved his job: "He really wanted to teach us things - sometimes we had debates".Â
The French presidency said a national tribute would be held for Mr Paty, and the hashtag #JeSuisSamuel (I am Samuel) began trending on social media, echoing the #JeSuisCharlie call for solidarity after the attack on Charlie Hebdo.
France has seen a wave of Islamist violence since the January 2015 attack, which left 12 people dead including famous cartoonists.Â
Three weeks ago, as the trial got under way, a man attacked and wounded two people outside the magazine's former offices.
How is France reacting?
Charlie Hebdo on Friday tweeted: "Intolerance just reached a new threshold and seems to stop at nothing to impose terror in our country."
Education Minister Jean-Michel Blanquer, who met leaders from teaching unions on Saturday, said in a recorded statement that Mr Paty had been killed by the "enemies of freedom" and France would "never back down when confronted by terror, intimidation."
In the National Assembly, deputies stood up to honour the teacher and condemn the "atrocious terror attack".
Muslim leaders in France also condemned the attack. "A civilisation does not kill an innocent person, barbarism does," Tareq Oubrou, imam of a mosque in Bordeaux, told France Inter.
Prime Minister Jean Castex tweeted that "our teachers will continue to awaken the critical spirit of the citizens of the republic, to emancipate them from all totalitarianism".Â
The Strasbourg-based Assembly of Chechens in Europe said in a statement: "Like all French people our community is horrified by this incident