Islamic State claimed responsibility for the deadly rampage along the city's most famous avenue, Las Ramblas, on Thursday which was packed with tourists taking an afternoon stroll.
The death toll could rise, with more than 100 injured, authorities said.
The injured and dead came from 24 different countries, the Catalan government said in a statement, ranging from France and Germany to Pakistan and the Philippines. Spanish media said several children were killed.
As security forces hunted for the van's driver, who was seen escaping on foot, police said they had killed five attackers on Thursday night in Cambrils, a town south of Barcelona, to thwart a separate attack using explosive belts.
Six civilians and a police officer were injured in Cambrils when the attackers ran them over in a car, before police shot them dead and carried out controlled explosions. Police said the Cambrils incident was linked to the van attack in Barcelona.
Before the van ploughed into the tree-lined walkway of Las Ramblas, one person was killed in an explosion in a house in a separate town southwest of Barcelona, police said. Residents there were preparing explosives, a police source added.
Police said they had arrested a Moroccan and a man from Spain's north African enclave of Melilla, though neither was the van driver. A third man was later arrested in the town of Ripoll in connection with the attack.
It was still not clear how many people had been involved in the van attack and other incidents on Thursday.
In Barcelona, witnesses said the white van zigzagged at high speed down Las Ramblas, a wide, busy avenue thronged with tourists, ramming pedestrians and cyclists, sending some hurtling through the air and leaving bodies strewn in its wake.
Police confirmed that another man who ran over two officers at a checkpoint in Barcelona on Thursday was shot dead, but it did not appear that the incident was linked to the van attack.
Islamic State's Amaq news agency said: "The perpetrators of the Barcelona attack are soldiers of the Islamic State and carried out the operation in response to calls for targeting coalition states" - a reference to a US-led coalition against the Sunni militant group.
Spain has several hundred soldiers in Iraq providing training to local forces in the fight against Islamic State, but they are not involved in ground operations. The Islamic State claim could not immediately be verified.
If the involvement of Islamist militants was confirmed, it would be the latest in a string of attacks in the past 13 months in which they have used vehicles to bring carnage to the streets of European cities.
That modus operandi - crude, deadly and very hard to prevent - has killed well over 100 people in Nice, Berlin, London and Stockholm.
An eyewitness to the Barcelona attack, Tom Gueller, told the BBC: "I heard screams and a bit of a crash and then I just saw the crowd parting and this van going full pelt down the middle of the Ramblas and I immediately knew that it was a terrorist attack or something like that.
"It wasn't slowing down at all. It was just going straight through the middle of the crowds."
Thursday's attack is the deadliest in Spain since March 2004, when Islamist militants placed bombs on commuter trains in Madrid, killing 191 people and wounding more than 1800.
Mobile phone footage showed several bodies strewn along the Ramblas, some motionless. Paramedics and bystanders bent over them, treating them and trying to comfort those still conscious.
Around them, the boulevard was deserted, covered in rubbish and abandoned objects including hats, flip-flops, bags and a pram.
"We saw a white van collide with people. We saw people going flying," holidaymaker Ellen Vercamm told El Pais newspaper.
An eyewitness named Rebecca told La Vanguardia: "I've seen a lot of people knocked down on the floor and the people are running and crying. The van drove down the middle of the street dragging everyone with it."
Belgium's foreign minister said a Belgian was among the dead.
Spanish Prime Minister Mariano Rajoy announced three days of official mourning for what he called a "jihadist attack."
The Spanish royal household said on Twitter: “They are murderers, nothing more than criminals who are not going to terrorise us. All of Spain is Barcelona.”
TOURIST DRAW
The attack in Barcelona took place at the height of the tourist season, and in one of Europe's top travel destinations, with at least 11 million visitors a year.
French President Emmanuel Macron, whose nation has suffered some of Europe's deadliest militant attacks in recent years, tweeted: "All my thoughts and France's solidarity to the victims of the tragic attack in Barcelona."
A Vatican spokesman said Pope Francis was praying for the victims and wanted to express his closeness to all Spanish people, especially the victims and their families.
US President Donald Trump said: "The United States condemns the terror attack in Barcelona, Spain, and will do whatever is necessary to help." He added: "Be tough & strong, we love you!"
Barcelona is the capital of the wealthy northeastern region of Catalonia, which plans to hold a popular vote on October 1 on whether it should secede from Spain. It is in dispute with the central government, which says the vote cannot go ahead because it is unconstitutional.
Following the attack, security staff at Barcelona airport suspended a strike that started in early August. "Our work is now more necessary than ever," a spokesman said.
