"Info available at this stage: from 2208 hrs (10:08 pm) officers responded to reports of a vehicle in collision with pedestrians on #London Bridge".
Police say the incidents occurred on the bridge and at Borough Market a short distance away.
British Transport Police said there were reports of "multiple" casualties following the major incident on the bridge.
The Telegraph has published reports of shots fired near The Shrad on Southwark Street, and have further reported that police on the scene told everyone to "run".
Police say they are also responding to an incident in the Vauxhall area.
Metropolitan Police said the van and market attacks are being considered terrorist incidents.
The BBC reports police are searching for three suspects who could be armed, according to Reuters.
"The United States stands ready to provide any assistance authorities in the United Kingdom may request", State Department spokeswoman Heather Nauert said.
The bridge was closed off in both directions, as was the London Bridge train station as well.
"A table was thrown, a bottle was thrown at the individual with the knife and then we heard three gunshots and we ran", he added.
"Van drove into people on London Bridge, this is happening now". US President Donald Trump has also been informed, his press secretary announced.
Witnesses said they saw a white van struck pedestrians on London Bridge.
"I saw that man with a knife in his hand and after that a man started screaming so I knew something wrong was happening", he said.
The current London Bridge opened in 1973 and is made of concrete and steel.
"He swerved right around me", Jones said, "and then hit about five or six people".
The patrons sheltered in the basement and police have arrived at the scene, the witness said.
"The police were absolutely brilliant, they came in and explained the situation", he said".
A married couple, Natalie and Ben, said they saw a man with "quite a large blade. maybe 10 inches".
"He hit about two people in front of me and then three behind".
The van swerved into oncoming lanes before hitting a bus stop and coming to a stop, Roberts said.
Trump also used the incident to promote his proposed travel ban, which has been blocked by federal courts.
One witness reported having seen a van crashed into a lamppost after the bridge by the Barrowboy and Banker pub.
A witness on the bridge told CNN the van mowed down pedestrians as it sped south across the bridge, leaving bodies lying in the roadway.
On March 22, the British Parliament building was attacked by a man who ran over pedestrians with a vehicle and then stabbed several people.