Police were there within a few minutes and the three men believed to be responsible for the attack were killed by the police.
"Following updates from police and security officials, I can confirm that the bad incident in London is being treated as a potential act of terrorism", Ms May said.
Bursts of gunfire echoed through the streets - likely from armed police - and at least three blasts rang out as police performed controlled explosions.
At roughly the same time, there was a knife attack in a London market.
Assistant Commissioner Mark Rowley of the Metropolitan Police said: "At this stage, we believe that six people have died in addition to the three attackers shot dead by police".
Trump second tweet provoked sharp criticism from commentators who saw it as a piece of opportunism at London's expense.
Police were called to London Bridge at 22.08pm.
"The Conservative Party will not be campaigning nationally today". Out of respect the SNP has suspended national campaigning this morning.
Prime Minister Theresa May said she was being updated by officials on the "fast-moving" situation, adding: "I can confirm that the awful incident in London is being treated as a potential act of terrorism".
In a statement, Mr Nuttall said: "With more people murdered on the streets of our capital city last night by Islamist terrorists, it is more important than ever for us to confront this evil with the democratic principles that have made this country what it is".
Indonesia too has been battling Muslim militants groups that have staged attacks across the sprawling archipelago, including in the capital Jakarta.
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France is still under a state of emergency after a string of Islamic extremist attacks.
German Chancellor Angela Merkel said news of the attacks struck her with "sadness and dismay".
The opposition Labour Party has not yet announced its plans for Sunday.
It also comes less than two weeks after 22 people, including seven children, were killed in a suicide bombing at the end of a pop concert at the Manchester Arena. The Foreign Ministry also tweeted its condemnation of the attack.
Witnesses described a white van careering into pedestrians near London Bridge and knocking over several people. They asked Londoners to "remain calm but be alert and vigilant".
He added: "I feel sick".
The prime minister said the election would be held as scheduled.
Scotland Yard has also confirmed that a third stabbing incident in Vauxhall was unrelated.
The assault began when a van veered off the road and barreled into pedestrians on London Bridge.
Canadian diplomats in England were advising Canadians to avoid the area around London Bridge and the nearby Borough Market until the areas were deemed safe.
More than 30 victims are being treated at London hospitals and a number of others suffered less serious injuries.
May, facing an election Thursday, said the recent attacks, including an earlier one outside Parliament in March, are not directly connected, but that "terrorism breeds terrorism" and attackers copy one another. "If I had fallen over they probably would have killed me".