Paris attacks: UK police working 'closely' with French
British security services are working "very closely" with French and Belgian authorities to find anyone involved in the Paris attacks, Theresa May says.
The home secretary warned people in Britain to be "alert but not alarmed" and said increased police patrols and border searches were taking place.
The UK stands "in solidarity" with France, she said, adding: "The terrorists will not win."
Concerns remain over a "handful" of Britons in Paris, she added.
One Briton - Nick Alexander from Essex - has been confirmed dead, and Mrs May said others had been injured.
Friday's attacks by gunmen and suicide bombers left about 129 dead and 350 injured.
A one-minute silence for the victims will be held in the UK at 11:00 GMT on Monday. It will be part of a Europe-wide silence at midday French time.
On Sunday prayers were said for the people of Paris at religious events across Scotland. Vigils were also held in Birmingham, Leeds and outside London's City Hall.
Mrs May chaired a Cobra emergency committee meeting about the lessons to be learned from the Paris attacks.
Setting out the British response, she said: "The UK police and security services are working very closely with their counterparts in France and Belgium to identify all those involved and to pursue anyone who may have been involved in the preparation of these barbaric attacks."
Asked if there was a security threat from Islamic State (IS) militants hiding among migrants coming to the UK, she told the BBC's Andrew Marr Show that Britain took only the most vulnerable people from Syrian refugee camps, and rigorous checks were in place.
She later said a "crisis team" had been sent from the Foreign Office to Paris to help British citizens who had been injured or witnessed the attacks.
Officers from the Metropolitan Police's Counter Terrorism Command unit are interviewing people returning to the UK from France who may have information about the Paris attacks.
The Met is also appealing for any potential witnesses to contact its anti-terrorist hotline, on 0800 789 321.
Prime Minister David Cameron said the UK's safety and security depended on "degrading and ultimately destroying" IS - or what the government calls ISIL - which has said it carried out the attacks.
Speaking from Turkey, where he is attending the G20 summit, he added: "We're playing a huge role in that already in Iraq.
"Others are taking action in Syria which we both support and enable, but we've got to keep on making the case that we will be safer in the UK, in France, right across Europe, if we destroy this death cult once and for all."
Mr Alexander, 36, from Colchester, died in the attack at the Bataclan concert hall, where he had been selling merchandise.
His family described him as "generous, funny and fiercely loyal".
Friends of Mr Alexander also paid tribute to him on social media. Joe Trohman, lead guitarist of the rock band Fall Out Boy, described him as a "great guy".
More than 80 people died at the concert hall, one of a number of locations targeted by marauding gunmen.
The US band Eagles of Death Metal were playing a gig when attackers burst into the venue and opened fire, but the band themselves survived unscathed.
Other bands, including U2, Motorhead and Foo Fighters, have cancelled gigs in Paris.
By Susana Mendonça, BBC political correspondent
The terror attacks in France may not have altered the threat level in Britain yet, but what's clear from the home secretary's remarks is that it's a fluid situation which could change as the French investigation unfolds.
The threat level has been set at "severe" in Britain for more than a year now, but since the shocking events in Paris she said there had been an increased police presence on Britain's streets and tighter border checks - with more freight vehicles being screened.
But the overriding message coming from Mrs May was one of reassurance and of solidarity.
She told Britons to be "alert but not alarmed" and - with a handful of possible British victims in the Paris attack - she reiterated that Britain would stand shoulder to should with France.
Michael O'Connor, from South Shields, was at the Bataclan concert hall when gunmen attacked.
The 30-year-old said he lay on top of his girlfriend to protect her as other audience members were shot and killed. Both escaped unharmed after playing dead.
"My girlfriend was so brave... she kept on telling me this isn't where it's going to end, we're not going to die here," he said.
Peter McNiven, from Guildford in Surrey, told the BBC his son Jack, who lives in Paris, was shot and injured at the bar Le Carillon, which gunmen attacked before targeting diners at a restaurant across the road.
"He had just ordered drinks from the bar and I think suddenly there were gunshots. He was hit, his friend was hit," he said.
Both Jack and his friend survived but were taken to hospital.
"When we spoke to him in the ambulance he kept saying 'I'm OK, there are lots of people much worse than me'," Mr McNiven said.