London's Metropolitan Police said the incident on the bridge occurred at around 10:08 p.m. (5:08 p.m. ET) on Saturday. Within eight minutes, three attackers had been shot dead.
At least 20 people were hospitalized.
Following the incident on the bridge, police were called to reports of a stabbing at adjacent Borough Market — which features many restaurants and pubs. Armed officers responded and shots were fired, police said.
At least 20 people were taken to six hospitals across the city, with others treated at the scenes as well, health officials said.
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London Mayor Sadiq Khan described the incident as a "horrific terrorist attack," adding: "This is a deliberate and cowardly attack on innocent Londoners and visitors to our city enjoying their Saturday night."
Gerard Vowls, 47, described witnessing people being stabbed, seeing blood, and realizing it wasn't a "joke."
He said: "I was going, 'Everybody, run — they're terrorists! terrorists! they're killing everybody!' and they're going, 'This is for Allah,' and they're just running around."
The attacks on London Bridge and at the market come two months after a driver plowed into pedestrians on Westminster Bridge in what authorities have called a terror attack, and almost two weeks after a deadly suicide bombing outside an Ariana Grande concert in Manchester.
Fabio Lamas was working at the Wheatsheaf Pub in Borough Market when he said he heard someone shout "knives, knives, knives" and he saw three people holding knives who appeared to be wearing body armor outside. A man moved to block the door, and people inside heard shooting outside, Lamas said.
Witnesses at another Borough Market restaurant said an attacker entered and stabbed a woman, and its manager and others threw bottles at the men and staff pushed them out outside using a bar stool.
Video posted on social media showed diners at one restaurant hiding in a cellar.
Railway worker Matt Cooper said he was going out for a coffee near Borough Market and "I saw loads of people running towards me and I heard gunfire." He said he heard about 12 shots. Police told everyone to run.
Two nearby hospitals were locked down to protect patients and staff.
President Donald Trump said on Twitter: "Whatever the United States can do to help out in London and the U. K., we will be there."
BBC reporter Holly Jones was on London Bridge at the time of the incident and said the van was driven by a man who appeared traveling around 50 miles per hour.
"He swerved right round me and then hit about five or six people. He hit about two people in front of me and then three behind," Jones said, the BBC reported. Jones confirmed the account in an interview with MSNBC.
Video on social media showed people being evacuated from the area. Witness Peter Joshuas told NBC News he was just exiting the bridge when he heard screams and people shouting and running in confusion.
Faith Buscemi, a security officer near London Bridge said, "I saw people running, screaming — somebody was injured. I see people with blood. It was the worst 30 minutes of my life, honestly."
NBC News producer Mo Abbas was at a restaurant near the south entrance to London Bridge, when the incident happened. He said a panicked store manager told diners to abandon their meals and shelter in the back of the restaurant while police conducted a search of the area.
After about 10 minutes, Abbas said diners were escorted single-file out of the restaurant and led away from the area by armed police in body armor.
Saphora Smith, Chloe Hubbard, Fiona Day, Jason Cumming and Mo Abbas reported from London. Phil Helsel reported from Los Angeles.