Police responded rapidly with dozens of emergency vehicles and authorities led shell-shocked members of the public away from the scene with their hands on their heads.
There were also reports of a third incident, a stabbing in the Vauxhall area of London.
Metropolitan Police said the van and market attacks are being considered terrorist incidents.
The BBC reports police are searching for three potentially armed suspects.
"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", May said in a statement.
London Ambulance have said: "We're responding to this incident and more information will follow when we have it". The drivers of the van then ran out with knives, attacking people on the street and in businesses in nearby Borough Market.
American president Donald Trump tweeted after the incident: "We need to be smart, vigilant and tough".
Lara Al-Ostta told The Guardian she was having drinks with a friend in a bar underneath London Bridge when people came running into the pub and telling everyone to run away because "people are stabbing each other". Multiple people have suffered serious injuries.
The attack began when a van traveling at a high speed deliberately crashed into pedestrians near London Bridge.
"He swerved right 'round me and then hit about five or six people".
The Daily Mail reports that at least seven people are believed to have been stabbed while eyewitnesses said bodies were "strewn" on the road.
He added: "I've been in situations like this before, I don't think you realise how scary how possibly be until you put yourself into it, a police man could have shot me because it was me, the police and these three men so I was lucky definitely".
Indians are also speculated to be in the area as there is an India vs Pakistan match scheduled at the Edgbaston Cricket Stadium in Birmingham on June 4.
After ramming the vehicle into a barrier outside the Houses of Parliament, the driver, 52-year-old Khalid Masood, got out and stabbed a police officer to death before being gunned down by responding officers. Five victims were killed in the attack.
Venus Williams opens play at French Open on Day 4 She came closest in 2002 when she was beaten in the final by her younger sister, as she was at both Wimbledon and the U.S. Kvitova double faulted into the net on match point and spiked her Wilson racquet in frustration at that transgression.
CBS News confirms that Special Counsel Robert Mueller has cleared former FBI Director James Comey to testify before Congress . A number of congressional committees, in addition to the Senate intelligence panel, are also looking into the matter.
Everyone has been following the creation and promotion of LaVar Ball's new athletic company " Big Baller Brand " pretty closely. Lonzo Ball averaged 14.6 points, 6.0 rebounds and 7.6 assists in his one-and-done season at UCLA.
All of Allman's five children except Island - each of the five has a different mother - have played music professionally. Cher , 71, and Gregg , 69, were married for four years before divorcing in 1979 and they have son Elijah Blue together.
He was in the process of negotiating peace with the Palestinians, who also had political claims to Jerusalem . The announcement postpones the president's campaign promise to move diplomatic operations to Jerusalem .
In the incident report filed by the four officers at the scene, Woods kept dozing off even after his initial contact with police. Woods had said on his website last week that he would not be able to twist his back for three months because of his surgery.
Is this the kind of person Kathy Griffin wants to be? 'And really, the comedians are the last voice of truth in this whole thing. Dave Berg, author of Behind the Curtain: An Insider's View of Jay Leno's Tonight Show, co-produced the show for 18 years.
Mr Trump, who BBC Europe editor Katya Adler said seemed star-struck, said of the Pope: "He is something, he's really good". But Pope Francis pulled a great move in his first meeting with the President of the United States.
Czech second seed Karolina Pliskova reached the French Open second round on Monday with a 7-5, 6-2 win overn China's Zheng Saisai. KEY STATISTIC I: 9 - Record number of French Open titles for Rafael Nadal , who is favored by many to collect No. 10 this year.
Daniel Ricciardo brought his Red Bull across in a distant third place to complete the podium, his best result of the season. Pulling out a small lead, Hamilton eventually began chipping away at the Ferrari's advantage.
When she was in jail, Mercedes brought her a puppy called Stanley - "one of the puffball dogs that old people usually have". ABC News reported that Schapelle Corby's home will receive extra police protection ahead of her long-awaited return home.
The Warriors , who beat Cavs for the 2015 championship before losing the rematch in 2016, are 12-0 in this year's playoffs. Cleveland has steamrolled through the East in the postseason since James returned home, going 36-5 in three years.
Neill Blomkamp Is Making Short Movies For Steam
These short films would be heavily monster focused, and the studio would serve as a testing ground for potential full-scale films. In the wake of his scrapped Alien sequel , Blomkamp has been teasing his new project, Oats Studios , for some time.
VanWynsberghe takes 41st at national bee
Daniel Doudna, 13, of Fairbanks, Alaska, said it was "amazing" to be the third child in his family to attend the national Bee. Jacques Bailly, questions about the word to gather information that might help her determine the correct spelling.
Trump pulls United States from Aust on Paris, trade
A voluntarily promise to reduce greenhouse gas emissions in an effort to combat climate change was agreed in 2015. Trump tweeted yesterday morning: "I will be announcing my decision on the Paris Accord over the next few days".
Philippine soldiers killed in 'friendly fire'
The minister raised the possibility of limiting airstrikes if government troops converge within cities controlled by militants. Fighters loyal to him surprised government forces with their firepower, fending off air strikes and house-to-house searches.
AMD Vega to be released in August!
Previous ROG gaming products based on AMD designs included Athlon 64 FX processors and were desktop models, until now. Speaking of Vega, AMD announced that it would launch the new graphics cards at the end of July at Siggraph 2017.
Dixon's eventful Indy includes pole, robbery and crash
Howard said he wasn't sure what caused him to hit the wall, whether a part broke in the suspension or something else. Takuma Sato , who won this year's race, became the first Japanese driver to win the Indy 500, The Drive reported .
BJP says it will give right position to Rajinikanth
His May 19 speech was replete with hints of political ambition while the superstar met with his fans after a long time. Or like the cigarette that he flicks in the air in a trademark style, the frenzied talk will just end up in smoke.
ICSE, ISC results to be announced today at 3 pm
In 1973, the Council was listed in the Delhi School Education Act 1973, as a body conducting "public" examinations. And so, after almost two weeks of delays and uncertain reports, ICSE results 2017 are finally coming today.
Philippines says foreigners among IS-linked gunmen in south
Mahmud, who is also known as Abu Handzalah, led fellow militants to rescue Hapilon in the Marawi attack, the sources say. He has been identified as a chief recruiter by the militants who trained and sent Malaysians to fight in Syria and Iraq.
US beats Kiwis on opening day of America's Cup qualifiers
If one of the finalists was also the victor the Louis Vuitton America's Cup Qualifiers, they will start with a one point bonus. Oracle and the five challengers will compete in two round robins, which will eliminate one challenger.
Trump pushes aside Montenegro leader _ who calls it natural
Trump promised to launch a review , and said, "Leaks of sensitive information pose a grave threat to our national security". Donald Trump backs the view that Brexit is an "incident not a trend", European Council President Donald Tusk has said.