- 2 attacks near British Parliament in London
- Policeman stabbed, attacker shot dead
- Car runs over people at Westminster Bridge
"The location of this attack was no accident," she said in a statement outside her 10 Downing Street office late in the evening.
"The terrorist chose to strike at the heart of our capital city, where people of all nationalities, religions and cultures come together to celebrate the values of liberty, democracy and freedom of speech."
Any attempt to defeat those values through violence was "doomed to failure", May said.
"A car then crashed near to parliament and at least one man, armed with a knife, continued the attack and tried to enter parliament," Rowley said.
He said the police's "fast-paced investigation" was working on the assumption that the attack was "Islamist-related terrorism". Police believed they knew the identity of the attacker but would not provide details at this stage, he said.
It was the deadliest attack in London since four British Islamists killed 52 commuters and themselves in suicide bombings on the city's transport system in July 2005, in London's worst peacetime attack.
It took place on the first anniversary of attacks by Islamist terrorists that killed 32 people in Brussels.
Reuters reporters inside parliament during Wednesday's attack heard loud bangs and shortly afterwards saw the knifeman and the stabbed policeman lying on the ground in a courtyard within the gates of parliament.
The dead police officer was identified as Keith Palmer, 48, with 15 years of service.
A Reuters photographer saw at least a dozen people injured on the bridge. His photographs showed people lying on the ground, some of them bleeding heavily and one under a bus.
A woman was pulled alive, but with serious injuries, from the Thames, the Port of London Authority said. The circumstances of her fall into the river were unknown.
Three French schoolchildren aged 15 or 16 were among those injured in the attack, French officials said.
Several members of parliament (MPs) and senior officials were caught up in the chaos. Tobias Ellwood, a junior Foreign Office minister, was pictured attempting to resuscitate a man lying unconscious, reported to be the stabbed policeman.
London Mayor Sadiq Khan said there would be additional police officers on the city streets to keep Londoners and visitors safe.
"We stand together in the face of those who seek to harm us and destroy our way of life. We always have, and we always will. Londoners will never be cowed by terrorism," he said.
Parliament's lower House of Commons, which was in session at the time, was suspended and lawmakers were asked to stay inside the building for several hours.
May said parliament would convene as normal on Thursday in a sign that the attack would not disrupt life in the capital.
However, a visit by Queen Elizabeth to officially open the new headquarters of London's police force, which had been planned for Thursday, was postponed "in light of today's events," Buckingham Palace said.
In a telephone call with May, President Donald Trump offered Britain the full cooperation and support of the United States, the White House said. US Secretary of State Rex Tillerson condemned the attack as "horrific acts of violence".
German Chancellor Angela Merkel and NATO Secretary General Jens Stoltenberg were among foreign leaders who expressed shock and solidarity.
Eyewitnesses described scenes of panic during the attack.
"I just saw a car go out of control and just go into pedestrians on the bridge," said Bernadette Kerrigan, who had been on a tour bus on the bridge, in an interview with Sky News.
"As we were going across the bridge, we saw people lying on the floor, they were obviously injured. I saw about 10 people maybe. And then the emergency services started to arrive. Everyone was just running everywhere."
Former Polish Foreign Minister Radoslaw Sikorski, who witnessed the attack from a taxi as he crossed the bridge, said he saw five victims and made a video of the scene.
"I heard something that sounded like a small car crash. Then I looked out of the window and saw that there was one person lying on the asphalt," he told Reuters.
"I did not see the face of the person lying on the asphalt, but the person was not moving, it was not showing any signs of life. One of the men I saw, his head was bleeding very badly. But the person I filmed -- no, that person was not showing any signs of life."
Journalist Quentin Letts of the Daily Mail newspaper told LBC radio that he had witnessed the stabbing of the policeman and the shooting of the assailant from his office in the parliament building.
"He (the assailant) ran in through the open gates ... He set about one of the policemen with what looked like a stick," Letts said.
In Edinburgh, the Scottish parliament suspended a planned debate and vote on independence as the news from London came in.
Britain is on its second-highest alert level of "severe" meaning an attack by militants is considered highly likely.
The most recent deadly attack in London to be treated as a terrorist incident was in May 2013, when two British Islamists stabbed to death soldier Lee Rigby on a street.
(Additional reporting by Kylie Maclellan, Elizabeth Piper, Costas Pitas, Alistair Smout, Michael Holden, Kate Holton, Elisabeth O'Leary, Andy Bruce, David Milliken and William Schomberg; Writing by Estelle Shirbon; Editing by Stephen Addison, Guy Faulconbridge, Giles Elgood, Catherine Evans and Jonathan Oatis)