On London Bridge, witnesses described seeing a white van mounting the sidewalk and injuring at least six people at about 10:30 p.m.
The service says "we have taken at least 20 patients to six hospitals across London".
Another witness, Will Heaven, said he saw people who appeared to have been hit, and one being put into an ambulance.
British Transport Police said there were reports of multiple casualties following the incident on the bridge, The Associated Press reports.
Asking people to avoid the area, British police said multiple resources were attending the incident.More news: Cristiano Ronaldo bags brace as Real Madrid clinch Champions League
The attacks come days ahead of an 8 June election and less than two weeks after a suicide bomber killed 22 people at a pop concert by USA singer Ariana Grande in Manchester in northern England.
Scotland Yard was looking for three men who reportedly arrived at London Bridge around 2210 GMT and stabbed people randomly. "He hit about two people in front of me and then three behind", he said.
Three civilians were killed on the bridge and one police officer was stabbed to death outside the parliament.
Shortly after the incidents, May said the "terrible" incident in London is being treated as a "potential act of terrorism", The Independent reported.
- Police have confirmed there has been more than one fatality.
London's Metropolitan Police force said it had declared the attacks as "terrorist incidents".More news: Scottish First Minister Extends Condolences Following Terror Incidents in London
In March, a knife-wielding man plowed a vehicle into pedestrians on London's Westminster Bridge and later stabbed a police officer to death inside the gates of Parliament.
The bridge was closed from both directions.
On March 22, Khalid Masood ploughed a rented vehicle into pedestrians and stabbed a policeman to death before being shot dead.
A White House spokesman said President Donald Trump has been briefed by his national security aides on the unfolding events in London, but that US officials do not have any details beyond media reports.
"Our thoughts (and) prayers are (with) the victims, courageous first responders (and) all the people of London", Pence wrote.More news: Vladimir Putin to Megyn Kelly: US Hackers Could Have Framed Russia
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