Though Prime Minister Theresa May initially said they were being treated as "potential acts of terrorism", according to The New York Times, only the incidents at London Bridge and Borough Market have been confirmed as acts of terrorism, according to the Metropolitan Police.
Police have confirmed they also "responded to an incident" in the nearby Borough Market after reports suggested there was a knife attack there.
According to The Telegraph, some 15-20 pedestrians on the London Bridge were struck by a white van after the vehicle, driving at about 50 miles per hour, appeared to intentionally jump a curb on the bridge at approximately 10:30 p.m. local time. Police further said armed officers were responding to a third incident in the Vauxhall area.
Police say a hit-and-run and stabbings on and near London Bridge have been officially declared a "terrorist incident".
The white van veered off the road on the bridge and hit five or six people on Saturday. Witness Peter Joshuas told NBC News he was just exiting the bridge when he heard screams and people shouting, and people running in confusion.
Witnesses reported seeing injured people on the ground on the bridge. "We ran for like 100 meters and then saw loads of police cars turned up and there was kind of a period of quite intense gunfire", he said.
Police say they are working with Royal Naval Lifeboat Institution to evacuate public from the London Bridge incident.
"I mean, of course there is panic, of course there is fear, of course there is chaos, but that being said I think amongst all of that the way in which the London police are handling this situation is a real testament".
London attack: What we know so far
London Bridge station is also closed. "He swerved right round me and then hit about five or six people". London's Metropolitan Police tweeted that a third incident has occurred in the Vauxhall area.
Another witness, Mark Roberts, told CNN he saw the van hit multiple people before hitting a bus stop and stopping.
People walk away from inside a police cordon after an incident in London.
Minutes earlier, he posted a tweet touting his immigration ban, although he did not specifically mention the London incidents.
The drivers reportedly got off the van to stab people in the nearby Borough Market area of bars and restaurants. May will chair a meeting of the government's emergency response committee on Sunday.
Two suspected terror incidents are happening in London this evening. Medics treated people lying injured on the ground near the market as shocked people cried and shouted around them. Witnesses reported seeing armed police fire into a Tapas restaurant at the market. At least one person has died from their injuries on the bridge.
London Ambulance Service said "multiple resources" were sent to the scene.
The Metropolitan Police advised people caught up in the incidents to "RUN - to a place of safety".
In the wake of the Manchester attack, the United Kingdom government increased the terror alert to critical, the highest level, and deployed the army to support police.
The incidents in London come less than two weeks after 22 people were killed in the northern city of Manchester when a suicide-bomber detonated explosives as fans were leaving a concert by pop singer Ariana Grande.