Ex-chancellor George Osborne was fired, the BBC`s Laura Kuenssberg said.
On arriving at Downing Street, Mrs May vowed to lead a government that works for all, not just the "privileged few".
The UK`s second female prime minister promised to give people who were "just managing" and "working around the clock" more control over their lives.
Downing Street said she took a number of "congratulatory phone calls" from other European leaders on Wednesday evening, where she told them "some time" would be needed to prepare for EU exit talks.
In her appointments announced so far, Michael Fallon continues as defence secretary, and Liam Fox, who resigned as defence secretary in 2011, has a new role as secretary of state for international trade.
Theresa May`s new cabinet appointments
Chancellor - Philip Hammond
Home Secretary - Amber Rudd
Foreign Secretary - Boris Johnson
Defence Secretary - Michael Fallon
Secretary of State for exiting the European Union - David Davis
International Trade Secretary - Liam Fox
Mrs May has spoken of the need to reunite the Conservative Party after the EU referendum, in which she backed a Remain vote.
Her new line-up includes big jobs for prominent Leave campaigners Mr Johnson - who was initially tipped as the next prime minister after the Brexit vote - Mr Fox, and Mr Davis, whose job title is "secretary of state for exiting the European Union".
UKIP leader Nigel Farage tweeted that the appointments of Mr Fox and Mr Davis were "inspired choices", adding: "I feel more optimistic now."
On Twitter, Mr Osborne, who was chancellor throughout Mr Cameron`s tenure, said the job had been a "privilege", adding: "Others will judge - I hope I`ve left the economy in a better state than I found it."
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The BBC`s political editor Laura Kuenssberg said Mr Osborne had been "told his services were no longer required" because his "brand" was seen as "too tarnished".
Mr Johnson said he was "very humbled" to be appointed foreign secretary.
He said Mrs May had made a "wonderful speech" earlier, saying there was a "massive opportunity in this country to make a great success of our new relationship with Europe and with the world".
But Lib Dem leader Tim Farron predicted Mr Johnson would "spend more time apologising to nations he`s offended" than working as foreign secretary.
The new prime minister had spoken by phone to German Chancellor Angela Merkel, French President Francois Hollande and Irish Taoiseach Enda Kenny, Downing Street said late on Wednesday.
A Number 10 spokeswoman said the PM had "emphasised her commitment to delivering the will of the British people to leave the European Union".
She added: "The prime minister explained that we would need some time to prepare for these negotiations and spoke of her hope that these could be conducted in a constructive and positive spirit."
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