The London ambulance service said at least 20 people had been taken to six hospitals across the city.
A witness, who was doing a night photography class on London Bridge, said he saw the van veering on and off the pavement on the bridge.
It also recalled an attack on Westminster Bridge when 52-year-old British Muslim convert Khalid Mahmood rammed his vehicle into pedestrians before crashing into the barriers surrounding parliament and then stabbing a police officer to death.
Police responding to multiple incidents near London Bridge. A Reuters reporter near the scene said she saw 10 police cars rushing towards London Bridge, Reuters reported.
There were also reports of a third incident, a stabbing in the Vauxhall area of London.
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A stabbing at Vauxhall was unrelated, the force added.
Police have reportedly arrested one man, and are searching for multiple suspects who may be armed.
Heaven also said he heard reports of stabbings. They also issued safety instructions to those in the areas of the incidents, warning people to run, hide, and contact police when it is safe to do so.
"I heard many gunshots and I heard people running away", Joe Dillon, 23, told the Washington Post.
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Witnesses on London Bridge reported seeing a van mounting the pavement and hitting pedestrians and a man with a knife running. "He hit about two people in front of me and then three behind", she said.
London Bridge has been closed down. Following the van attack, witnesses say at least three men spilled from the vehicle, carrying large knives.
"A police officer was trying to put himself between them and the crowd".
A cab driver said he saw three men with "twelve inch" blades get out of the van after the crash and start "stabbing people at random" on Borough High Street. Jones said the van struck five or six people, and that she witnessed a man later being taken away in handcuffs.
There are reports of a number of casualties, British Transport Police has said.
In March, four people were killed in London after Khalid Masood rammed his vehicle into pedestrians on Westminster Bridge, before fatally stabbing a policeman outside the Houses of Parliament. Five victims were killed in the attack.
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