Boris Johnson said he wanted to “change the country for the better” after he became the UK’s new prime minister.
Speaking outside Downing Street, he said the UK would leave the EU on 31 October “no ifs, no buts”, adding: “The buck stops with me.”
“The doubters, the doomsters, the gloomsters” who said it could not be done were “wrong”, the new PM added.
Mr Johnson is making sweeping changes to his ministerial team, sacking a host of big cabinet names.
Penny Mordaunt, Liam Fox and Greg Clark have all been removed from their posts – all three supported Jeremy Hunt, Mr Johnson’s rival, in the Tory leadership contest.
Education Secretary Damian Hinds has also gone, along with Chris Grayling, whose record as Transport Secretary has been much criticised.
In a 13-minute speech outside Downing Street, Mr Johnson listed a wide range of domestic ambitions, chiefly a promise to sort out care for the elderly “once and for all”.
Reforms to the social care sector have eluded previous governments because of their cost and complexity.
“We will fix it once and for all with a clear plan we have prepared to give every older person the dignity and security they deserve,” he insisted.
Mr Johnson also pledged to improve infrastructure, recruit 20,000 new police officers and “level up” school spending. He promised reforms to ensure the £20bn in extra funding earmarked for the NHS “really gets to the front line”.
And he pledged to boost the UK’s biotech and space science sectors, change the tax rules to provide incentives for investment, and do more to promote the welfare of animals.
The new PM will shortly begin announcing some senior cabinet appointments, and is currently meeting ministers whom he plans to dismiss at his offices in Parliament.
In a surprise move, Defence Secretary Penny Mordaunt, a leading Brexiteer who is popular across the party, has been sacked.
Moments later it emerged that another Brexiteer, Liam Fox, had also been removed from his post.
And Business Secretary Greg Clark, a vocal opponent of a no-deal Brexit, has also left.
There has been speculation that Mr Hunt, whose own future is uncertain, had been offered Ms Mordaunt’s job and turned it down.
Sweeping changes are expected after Chancellor Philip Hammond, Justice Secretary David Gauke and Cabinet Office minister David Lidington all quit in the past few hours.
Setting out his priorities for office, the former London mayor hit out at the “pessimists” who did not believe Brexit could be delivered and called for an end to three years of indecision.
“The people who bet against Britain are going to lose their shirts because we are going to restore trust in our democracy,” he said.
“The time has come to act, to take decisions and change this country for the better.”
He said he had “every confidence” the UK would leave the EU in 99 days time with a deal, but preparations for the “remote possibility” of a no-deal Brexit would be accelerated.
Mr Johnson vowed to bring all four nations of the United Kingdom – or what he described as the “awesome foursome” – together in the task of strengthening a post-Brexit country.
“Though I am today building a great team of men and women, I will take personal responsibility for the change I want to see,” he concluded.
“Never mind the backstop, the buck stops here.”
The BBC’s Vicki Young said she was struck by the ambition of Mr Johnson’s objectives beyond Brexit and the fact that he would take personal responsibility for his success or failure in achieving them.
Labour’s Sir Keir Starmer said Mr Johnson’s speech was “all rhetoric” and the new PM needed to show leadership rather than the “glib” answers he had become known for.
The new Lib Dem leader, Jo Swinson, said she would welcome a cross-party push to find a solution on social care, but attacked Mr Johnson’s “bluster and bravado” over Brexit.
Mr Johnson took over after Theresa May handed in her resignation to the Queen. A number of her senior ministers have also already resigned, saying they could not serve under her successor.
Earlier, as she relinquished power after three years, Mrs May said being prime minister had been “the greatest honour” and wished her successor well.
In a farewell speech outside No 10, she said his government’s “successes will be our country’s successes”.
Mr Johnson’s audience with Queen Elizabeth II lasted more than half an hour.
During his journey to Buckingham Palace, his car was briefly held up by protesters from Greenpeace, who formed a human chain across The Mall.
Mr Johnson’s partner, Carrie Symonds, and key members of his staff were awaiting the new prime minister’s arrival in Downing Street.