A number of Britons are feared to be among the dead following the terror in Paris.
Islamic State (IS) has claimed responsibility for the attacks which killed at least 127 people and wounded 300 more, around 80 of them critically.
David Cameron warned that the UK should "be prepared for a number of British casualties" as he vowed to the French people "your fight is our fight".
Speaking in Downing Street, he said the terror threat level in the UK would remain at "severe", but the attack would prompt a review of plans and suggested the threat posed by Islamic State was "evolving".
The Prime Minister said: "The events in Paris are the worst acts of violence in France since the Second World War, the worst terrorist attack in Europe for a decade, a horrifying and sickening attack.
"Our hearts go out to the French people and to all those who lost loved ones.
"Today the British and French peoples stand together as we have so often before in our history when confronted by evil."
Mr Cameron said the full picture of the attacks was still emerging but "we must be prepared for a number of British casualties and we are doing all we can to help those caught up in the attack".
He added: "These were innocent victims enjoying a Friday night out with friends and family, no doubt at the end of a hard week. They were not seeking to harm anyone, they were simply going about their way of life - our way of life.
"They were killed and injured by brutal and callous murderers who want to destroy everything our two countries stand for: peace, tolerance, liberty. But we will not let them.
"We will redouble our efforts to wipe out this poisonous, extremist ideology."
The Queen sent her condolences to the French people in a message to the president, telling Mr Hollande that she and the Duke of Edinburgh were "deeply shocked and saddened by the terrible loss of life".
A manhunt is under way for possible accomplices of gunmen who targeted a concert hall and the French national football stadium, and sprayed the terraces of bars and restaurants with gunfire in at least six almost simultaneous attacks.
In a message issued in French and Arabic, IS - also known as Isis, Isil and Daesh - said it had targeted "the capital of abominations and perversions and those who carry the crusader banner in Europe".
In an apparent reference to France's participation in airstrikes against IS strongholds in Syria, the statement warned the country, and others taking similar action, would "remain the principal targets of Islamic State ... for having struck Muslims in the territory of the caliphate with their aircraft".
French President Francois Hollande denounced the attacks as an "act of war" and vowed that France would be "merciless" in its response to the "absolute barbarity".
The French army and security forces were mobilised "at the highest possible level" and France would "act by all means anywhere, inside or outside the country" against the terrorists responsible, he said.
Speaking after an emergency meeting of senior government and security officials at the Elysee Palace, Mr Hollande declared three days of national mourning.
In a night of carnage in the French capital:
:: Police stormed the Bataclan concert hall where rock fans were being held hostage, but attackers wearing suicide belts blew themselves up, leaving 80 people feared dead. A witness said one of the gunmen shouted "Allahu Akbar" and said "This is for Syria".
:: Two suicide attacks and a bombing took place at the Stade de France stadium, where Mr Hollande was among thousands of football fans watching the national side play a friendly against Germany.
:: Gunmen targeted bars and restaurants in the 10th and 11th arrondissements of central Paris.
:: As many as 18 people died when the terrace of La Belle Equipe was sprayed with gunfire, while around 14 people were killed at Le Carillon bar-cafe. There were also shootings at the nearby Cambodian restaurant Le Petit Cambodge and the La Casa Nostra pizzeria.
The IS statement said that "eight brothers wearing explosive belts and carrying assault rifles" had struck targets "meticulously selected in advance in the heart of the French capital", including a football match between "crusader nations" and a "festival of perversity" at the Bataclan.
And the group warned that the attacks were "only the start of the storm".
Reports indicated that one of the Bataclan terrorists had been identified by his fingerprints as a French national.
A state of emergency was declared in France, with controls reimposed on the country's borders, police leave cancelled and 1,500 extra soldiers mobilised to guard official buildings and religious sites.
The Eiffel Tower has been closed following the attack, along with public facilities including schools. All public demonstrations are banned in the Paris region until Thursday.
Disneyland Paris was closed for the day and all sporting events in the French capital have been postponed, including a rugby union Champions Cup tie involving Scotland's Glasgow Warriors.
But the French football team's friendly against England at Wembley will go ahead on Tuesday after fears it could be cancelled following the violence in Paris.
Britons arriving back in London on the Eurostar told of their experiences during the terrorist rampage.
A group of students from Riddlesdown Collegiate in Croydon were on a school trip to the city when the attacks started.
Pupils and teachers sobbed as they arrived back at St Pancras Station amid tight security at the terminal.
"It's been obviously very worrying for us and for our parents," said principal Soumick Dey. "It's lovely to see them all safe back here now."