Two other incidents also were reported, stabbings at nearby Borough Market and another incident in the Vauxhall area of the city.
Police say they are dealing with an incident on the bridge but gave no further details.
British Transport Police said there were reports of "multiple" casualties following the major incident on the bridge.
Several witnesses also reported hearing gunshots.
Prime Minister Theresa May said it was a potential act of terrorism and the Metropolitan Police force said it had declared the attacks as "terrorist incidents".
The BBC reports police are searching for three potentially armed suspects. Paige Jegermanis, a 22-year-old American living in London, said soon after the bridge episode, a bartender informed the crowd in a riverside pub: "We're all on lockdown".
Police have said a third incident is developing in the Vauxhall area, which is about 2 miles southwest of the London Bridge area.
Trains are now not stopping at London Bridge or Borough stations.
Police were called to London Bridge at 22.08pm.
Several pedestrians were run over by a van in London on Saturday, according to multiple eyewitness reports.
VA holds annual Memorial Day ceremony at Wood National Cemetery
That leaves it to people like Resh, the Gold Star mother, to spread the message. Enjoy the day off if you have one, and remember the true roots of the day.
One told the BBC a white van veered off the road and struck as many as six people.
"He swerved right round me and then hit about five or six people", Jones continued.
We understand from Holly Jones that four pedestrians appear to be severely injured on London Bridge and that the van was going about 50 miles an hour, which is very unusual.
Ms Jones said police were on the scene within two minutes, quickly followed by ambulances.
Reports say more than a dozen people were injured.
She will hold an emergency ministerial meeting later on Sunday and Facebook has activated its safety check function for people in London to let their loved ones know they are safe.
The New York Police Department's Counterterrorism Bureau said on Twitter that its critical response teams had been deployed to heavily traveled pedestrian areas.
"They were right near the edge of the bridge". The police - the London Metropolitan Police - they're the most authoritative source for following these sorts of events in the city.
At Borough market, which in the evenings is busy with bars, witnesses said they saw a man with a large blade.
Witness Nick Archer told Sky News that he had seen a man lying on the ground who "looked in a bad way" and managed to flag down one of the police vans "whizzing past".