Police and ambulances were alerted to reports a van hit people on London Bridge shortly after 22:00 BST.
Armed officers went to nearby Borough Market where they were responding to reports of stabbings in the area which is known for its restaurants and bars.
London Ambulance Service said at least 20 people have been taken to hospital.
A number of people were treated at the scene for less serious injuries.
The Metropolitan Police tweeted: "At 0025hrs 4/6/17 the incidents at #LondonBridge & #BoroughMarket were declared as terrorist incidents."
Prime Minister Theresa May described the incidents as "dreadful events" and will chair a meeting of the government's Cobra emergency committee later.
Home Secretary Amber Rudd said the attacks were "horrific" and "targeted at people enjoying their evening with friends and family".
London Mayor Sadiq Khan said: "We don't yet know the full details, but this was a deliberate and cowardly attack on innocent Londoners and visitors to our city enjoying their Saturday night. I condemn it in the strongest possible terms."
Eyewitnesses told BBC home affairs correspondent Tom Symonds they saw between two and four people getting out of the van after the attack on the bridge and running towards Borough Market.
Gunshots were later heard in the area while one social media user has posted a photograph appearing to show one of the attackers lying on the ground outside a pub in the food market, with what seemed to be canisters strapped to his body.
The attack comes almost two weeks after 22 people died in a suicide bombing at a concert in Manchester.
In March, five people died in London when a car was driven at pedestrians on Westminster Bridge and a police officer was stabbed outside Parliament.
'Van hit five people'
The BBC's Tom Symonds said a security guard who oversees a number of pubs in the London Bridge area said he saw four people stabbed by three attackers.
BBC reporter Holly Jones, who was on the bridge at the time of the attack, said a van driven by a man was "probably travelling at about 50 miles an hour" before it hit a number of people.
"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," Ms Jones told the BBC News Channel.
Five or six people were being treated for injuries after the vehicle mounted the pavement and hit them, she said.
"I'd say there are about four severely injured people. They all have paramedics assisting them at the moment."
She said the van came from the direction of central London and headed towards the south side of the river.
Ms Jones later reported seeing a man being arrested by police. She said he was handcuffed and had his shirt off.
She said a French woman was among the injured. She had said she did not know where two people she had been with were.
Speaking to the Press Association, Will Orton described being in a pub in the area and seeing people coming running inside.
"We didn't really know what was going on," he said.
"We thought maybe there was a fight or something outside. And then there were almost hundreds of people coming inside.
"The bouncers did a really good job, they shut the doors and locked everyone in. There was panic - it seemed like it was literally outside the door. People were coming inside and saying they had witnessed people being stabbed."
The prime minister said: "Following updates from police and security officials, I can confirm that the terrible incident in London is being treated as a potential act of terrorism."
Mrs May added: "This is a fast-moving investigation. I want to express my huge gratitude to the police and emergency services who are on the scene.
"Our thoughts are with those who are caught up in these dreadful events."
Downing Street said Mrs May, who has been campaigning in the general election, returned to No 10 to receive further briefings from security officials.
On Twitter, Labour Party leader Jeremy Corbyn described the incidents as "brutal and shocking".
Transport for London (TfL) said London Bridge has been closed in both directions, while neighbouring Southwark Bridge has also been shut.
TfL said there were further closures in Borough High Street, where armed police and counter-terrorism specialist forces were later seen. On the north side of the river, road closures included Lower Thames Street and part of Cannon Street.
The Met Police has set up a casualty bureau on 0800 096 1233 and 020 7158 0197 for people concerned about friends or relatives who may be affected.
Meanwhile, Facebook said it has activated its safety check so people in London could post a message to let friends and relatives know they were safe.
Police said earlier they were also responding to an incident in the Vauxhall area a few miles away but later confirmed it was not connected to the London Bridge attacks.