David Cameron, who resigned as prime minister on Wednesday, has condemned the "sickening and dreadful" attack in Nice.
A sickening and dreadful attack in Nice. I know we stand with the French people and share their values. They shall never defeat us.
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It's been 37 years since Margaret Thatcher walked in to 10 Downing Street, but Theresa May is set to make her own mark on history.
Mrs May looks set to appoint a record number of women in her Cabinet later this week.
ITV News correspondent Juliet Bremner reports:
Theresa May moved into Number 10 on a momentous day in British politics, which saw David Cameron and his family leave Downing Street for the final time after six years in office.
Mrs May didn't delay getting straight to work, making key appointments in her new cabinet including Boris Johnson as Foreign Secretary.
ITV News' Political Editor Robert Peston reports:
Labour leader Jeremy Corbyn has congratulated Theresa May on becoming prime minister, but called on the new government to "abandon the destructive austerity policies which have damaged our economy and undermined living standards for most people".
The embattled Labour leader said in a statement: "I welcome Theresa May's acknowledgement that, after six years of Tory government in which she was a senior minister, the economy is not working for working people."
He added that "Labour will hold her government to account and make the case for a complete change of economic direction" and that the EU referendum result shows many in Britain feel left behind by globalisation and "it is essential that the new government recognises that and acts to ensure future generations have the chance to fulfil their potential".
"Labour is clear that democratic legitimacy for Theresa May's government can only come from a general election", the statement ends.
French President Francois Hollande has urged Theresa May to begin Brexit negotiations as quickly as possible, his office said after a phone call between the two leaders on Wednesday.
In a statement, the President's office said: "They agreed to actively develop the bilateral relationship that warmly unites France and Britain in all fields.
"The president repeated his desire that negotiations for Britain's exit from the European Union should be launched as quickly as possible."
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The prime ministers of Canada and Belgium have joined leaders from around the world in congratulating Theresa May on becoming prime minister.
Justin Trudeau, the Canadian prime minister, issued a statement congratulating Mrs May and paying tribute to David Cameron.
Meanwhile, the prime minister of Belgium, Charles Michel, took to Twitter to congratulate Mrs May, saying he hopes the UK will be a "close partner" of Belgium.
Russian President Vladimir Putin has thanked David Cameron for his time as UK prime minister, and congratulated Theresa May on replacing him.