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Boris Johnson made foreign secretary by Theresa May
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.
In the 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."
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".
Lib Dem leader Tim Farron predicted Mr Johnson would "spend more time apologising to nations he's offended" than working as foreign secretary.
After returning from Buckingham Palace, where she was formally appointed as prime minister by the Queen, Mrs May made a speech from outside 10 Downing Street.
Her husband Philip was standing behind her as she highlighted the "precious bond" between England, Scotland, Wales and Northern Ireland and between "every one of us".
"That means fighting against the burning injustice that if you're born poor you will die on average nine years earlier than others," she said.
For an "ordinary working class family", she added, "Life is much harder than many people in Westminster realise."
"When it comes to opportunity, we won't entrench the advantages of the fortunate few, we will do everything we can to help anybody, whatever your background, to go as far as your talents will take you," Mrs May said.
Mrs May also paid tribute to her predecessor, David Cameron, saying he had been "a great modern prime minister".
Mr Cameron earlier gave his final speech as prime minister, saying it had been "the greatest honour" of his life and that the UK was "much stronger" than when he took over.