The man believed to have carried out the attack in Westminster has been named by police as Khalid Masood.
Kent-born Masood, who died in the attack, was not the subject of any current police investigations, but had a range of previous convictions.
The 52-year-old was believed to have been living in the West Midlands.
The so-called Islamic State group has said it was behind the attack, in which PC Keith Palmer, Aysha Frade and US tourist Kurt Cochran were killed.
Eight arrests were made in London and Birmingham following Wednesday's attack.
Masood drove into pedestrians on Westminster Bridge before crashing his car into railings and then running into the grounds of Parliament, armed with a knife. He stabbed PC Palmer and then was shot dead.
Police said there had been no prior intelligence about his intention to carry out an attack.
But he was known to the police and his previous convictions included GBH, possession of offensive weapons and public order offences.
His first conviction was in November 1983 for criminal damage and his last conviction was in December 2003 for possession of a knife.
He had not been convicted of any terrorism offences.
Car hire company Enterprise said the vehicle used in the attack had been rented from its Spring Hill depot in Birmingham.
Masood had hired the Hyundai SUV in person, giving his profession as a teacher.
A JustGiving page set up for the family of PC Palmer reached its target of £100,000 on Thursday afternoon, less than 24 hours after it had been set up. That target has since been doubled.
Metropolitan Police said that as a mark of respect, the constable's shoulder number, 4157U, will be retired and not reissued to any other officer.
Mrs Frade worked at a London college, while Mr Cochran was from Utah, in the US, and had been visiting the capital with his wife Melissa, who is in hospital with serious injuries.
According to a family statement, the couple had been celebrating their 25th wedding anniversary and were due to return to the US on Thursday.
Mrs Frade and Mr Cochran were killed on Westminster Bridge.
Speaking in Parliament, Prime Minister Theresa May said the attacker was investigated some years ago over violent extremism but had been a "peripheral figure".
"He was not part of the current intelligence picture," she said.
May paid tribute to Palmer: "He was every inch a hero and his actions will never be forgotten."
She told MPs, many of whom had been caught up in the commotion: "We will never waver in the face of terrorism."
Rowley said hundreds of detectives worked through the night and searched six addresses, the BBC reported. Inquiries were held in London, Birmingham and other parts of the country.
"This attacker was inspired by international terrorism ... Islamist terrorism," Rowley said.
May said 12 Britons were admitted to hospital. Other victims included four South Koreans, three French children, two Romanians and Greeks each and one each German, Pole, Irish, Chinese, Italian and American.
One woman fell into the Thames from the bridge but was alive.
Indian President Pranab Mukherjee and Prime Minister Narendra Modi on Thursday joined world leaders to denounce the killings.
Mukherjee said India stood by the United Kingdom and that "terrorism must be met through collective action by the international community".
Modi added: "Deeply saddened... Our thoughts and prayers are with the victims and their families. At this difficult moment India stands with the UK in the fight against terrorism."
Flags were lowered to half mast over Downing Street.
The US, France and the European Union strongly condemned the attack.
President Donald Trump spoke with Theresa May and applauded the quick response of British authorities.
Witness Jayne Wilkinson said: "We were taking photos of Big Ben and we saw all the people running towards us. Then there was an Asian guy in about his 40s carrying a knife about seven or eight inches long.
"And then there were three shots fired, and then we crossed the road and looked over. The man was on the floor with blood.
According to Rowley, more police officers would be deployed on the streets of London. Officers' leave have been cancelled and duty hours extended.
Defence Secretary Sir Michael Fallon said the security at Parliament would be reviewed.
MPs held a minute's silence before Parliament continued business as normal.
Asked about the mood of the city, Fallon said: "London is getting back to normal. They've seen terrorism like this before and they are not going to let it triumph."
NO CHANGE IN CANADA'S THREAT LEVEL AFTER ASSUMED TERROR ATTACK IN LONDON
Public Safety Minister Ralph Goodale says Canada stands ready to help Britain in any way it can after what's believed to be a terrorist incident left at least four dead in London.
London police say four people have died, including an attacker and a police officer, and some 20 more have been wounded in the incident, which occurred on the grounds of the Parliament Buildings and the nearby Westminster Bridge.
Parliament remains in a lock-down while the police continue to search the area to ensure there are no other attackers.
Goodale says he's spoken with British Home Secretary Amber Rudd to offer Canada's assistance.
He says there's no change in the threat level in Canada.
But he says Canadian security agencies are taking all appropriate steps.
