The attack took place about 10:30pm local time, after a fireworks display. French president Francois Hollande said it was clearly terrorist in nature.
The president of the Nice region, Christian Estrosi, said the truck was loaded with weapons and grenades, and the driver had opened fire on the crowd.
"An individual drove a truck into the crowd. He was killed by police," said interior ministry spokesman Pierre-Henry Brandet.
A photograph showed the front of the truck riddled with bullet holes, and badly damaged with burst tyres.
Who was responsible?
Nothing is known about the attacker or their motives at the moment, but Mr Hollande said it was "attack of which the terrorist character cannot be denied".
He said that, following attacks last year in Paris, "all of France is under the threat of Islamic terrorism".
An interior ministry spokesman said: "Investigations are currently under way to establish if the individual acted alone or if he had accomplices who might have fled."
He denied reports that a hostage-taking incident had taken place.
'Like skittles in its path'
Damien German, a reporter with Nice Matin, described seeing the truck travelling at "breakneck speed" and manoeuvring in a manner so as to "mow down a maximum of people".
"I saw bodies like skittles in its path, heard noises, screams that I will never forget," he wrote.
"I was paralysed, I did not move... Around me there was panic. People were running, screaming, crying."
German described fellow Bastille Day revellers coming to help, bringing water and towels to those injured.
'We turned and ran'
Australian Emily Watkins, who was in Nice and joined the Bastille Day celebrations, said that while she saw the truck, she did not immediately realise what had happened.
"There was a lot of screams coming from ahead of us where the truck was and people just running towards us," she said.
"Without really knowing what was going on, we turned and ran as well.
"People were tripping over and trying to get into hotel lobbies and restaurants or car parks or anywhere they could to get away from the street."
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'We turned and ran'
Australian Emily Watkins, who was in Nice and joined the Bastille Day celebrations, said that while she saw the truck, she did not immediately realise what had happened.
"There was a lot of screams coming from ahead of us where the truck was and people just running towards us," she said.
"Without really knowing what was going on, we turned and ran as well.
"People were tripping over and trying to get into hotel lobbies and restaurants or car parks or anywhere they could to get away from the street."
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'There was a lot of panic'
ABC producer David Coady was among the crowd fleeing the scene in panic and said he could hear screaming and loud bangs.
"People were tripping over in the commotion, there was a lot of panic," he said.
"People were trying to get into hotels, any businesses that were open, trying to take shelter, because it was unclear what was happening.
"With each bang that we heard behind us, people perhaps started to go a bit faster, people were tripping over, it was a very chaotic scene."