Armed police rushed to the scene and shot dead the three male attackers in the Borough Market area near the bridge, as authorities urged Londoners on Twitter to "run, hide, tell" if they were caught up in the violence.
The attacks come days ahead of a June 8 election and less than two weeks after a suicide bomber killed 22 people at a pop concert by US singer Ariana Grande in Manchester in northern England. There was no immediate claim of responsibility.
"Sadly, six people have died in addition to the three attackers shot by police," Mark Rowley, Britain's top anti-terrorism officer, said before the death toll rose to 7. He said the three attackers had been wearing what looked like explosive vests that were later found to have been hoaxes.
The BBC showed a photograph of two possible London attackers shot by police, one of whom had canisters strapped to his body. Hours after the attack the area remained sealed off and patrolled by armed police and counter-terrorism officers.
The London ambulance service said more than 30 people had been taken to hospitals across the city. Three major London hospitals were on lockdown to keep patients and staff safe.
A Reuters reporter said some time after the attack began that he had heard loud bangs near the Borough Market area, but these appeared to be controlled explosions by security services.
Streets around London Bridge and Borough Market, fashionable districts packed with bars and restaurants, would have been busy with people on a Saturday night out.
Witnesses described a white van careering into pedestrians near London Bridge and knocking over several people.
"It looked like he was aiming for groups of people. I froze because I didn't know what to do," Mark Roberts, 53, a management consultant, told Reuters. He saw at least six people on the ground after the van veered on and off the pavement.
"It was horrendous," he said.
A taxi driver told the BBC that three men got out of the van with long knives and "went randomly along Borough High Street stabbing people."
Witnesses said people ran into a bar to seek shelter.
"People started running and screaming, and the van crashes into the railing behind. We went towards Borough Market and everyone went inside (the bar). Everyone in the bar started pushing people from the exits," one witness who gave his name as Brian, 32, told Reuters.
Another witness, who declined to be named, his white top covered in blood, described a scene of panic in the bar.
"They hit the emergency alarm. There was a line of people going down to the emergency exit. And then people started screaming coming back up," the 31-year-old said.
"Around the corner there was a guy with a stab wound on his neck ... There was a doctor in the pub and she helped him. They put pressure on the stab wound."
BBC radio said witnesses saw people throwing tables and chairs at the perpetrators of the attack to protect themselves.
ISIS calls for attacks
The BBC showed dozens of people, evidently caught up in the attack, being escorted through a police cordon with their hands on their heads.
ISIS, losing territory in Syria and Iraq to an advance backed by a US-led coalition, sent out a call on instant messaging service Telegram early on Saturday urging its followers to launch attacks with trucks, knives and guns against "Crusaders" during the Muslim holy month of Ramadan.
Similar attacks, in Berlin, Nice, Brussels and Paris, have been carried out by militants over the past two years.
"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," Prime Minister Theresa May said.
London's Thames river police said it worked with the lifeboat rescue service to help evacuate people caught up in the attack, described by police as a terrorist incident.
US President Donald Trump took to Twitter to offer US help to Britain. The White House said he had been briefed on the incidents by his national security team.
The incident bore similarities to a March attack on Westminster Bridge, west of London Bridge, in which a man killed five people after driving into a crowd of pedestrians before stabbing a police officer in the grounds of parliament.
Several witnesses also reported hearing gunshots around London Bridge.
"We were in an Uber (taxi) going towards London Bridge and suddenly we saw people running. The Uber stopped, we asked people what was going on, people said there was shooting," said Yoann Belmere, 40, a French banker living in London.
"Now the area is completely closed with police cars going one way and ambulances going the other," he told Reuters.
A witness told CNN two men had entered a restaurant in the Borough Market area and stabbed two people inside. He said a waitress was stabbed in the throat and a man was stabbed in the back.
The Manchester bombing on May 22 was the deadliest attack in Britain since July 2005, when four British Muslim suicide bombers killed 52 people in coordinated attacks on London's transport network.