Boris Johnson used his first speech as Prime Minister to insist that Brexit will be delivered, and that he will give the country "the leadership it deserves".
Speaking outside 10 Downing Street, the Tory leader said he would meet the October 31 deadline "no ifs or buts".
Watched by girlfriend Carrie Symonds, Mr Johnson promised he would "change this country for the better".
Arriving in Downing Street after being invited by the Queen to form a government during an audience at Buckingham Palace, Mr Johnson vowed to prove the Brexit doubters wrong.
He said: "I am standing before you today, to tell you the British people, that those critics are wrong – the doubters, the doomsters, the gloomsters are going to get it wrong again."
He predicted that "the people who bet against Britain are going to lose their shirts because we are going to restore trust in our democracy".
He added: "And we are going to fulfil the repeated promises of Parliament to the people and come out of the EU on October 31, no ifs or buts.
"And we will do a new deal, a better deal that will maximise the opportunities of Brexit while allowing us to develop a new and exciting partnership with the rest of Europe based on free trade and mutual support.
"I have every confidence that in 99 days' time we will have cracked it.
"But you know what we aren't going to wait 99 days, because the British people have had enough of waiting.
"The time has come to act, to take decisions, to give strong leadership and to change this country for the better."
He promised action to fix the social care crisis, make the streets safe and improve the NHS.
He said: "I will take personal responsibility for the change I want to see.
"Never mind the backstop, the buck stops here."
But on the issue of the Irish border – the main stumbling block in reaching a Brexit deal – Mr Johnson said he was "convinced" a solution could be found without checks at the Irish border and without the "anti-democratic backstop".
He added: "It is of course vital at the same time that we prepare for the remote possibility that Brussels refuses any further to negotiate and we are forced to come out with no-deal.
"Not because we want that outcome, of course not, but because it is only common sense to prepare."
Mrs May had used her farewell address in Downing Street to urge Mr Johnson to secure a Brexit deal.
She said the "immediate priority" was "to complete our exit from the European Union in a way that works for the whole United Kingdom".
The handover of power came on a dramatic day at Westminster which saw:
– Greenpeace protesters attempt to block Mr Johnson's car on its way to Buckingham Palace
– Cabinet ministers Philip Hammond, David Gauke and Rory Stewart hand in their resignations to Mrs May before she left office
– European Council president Donald Tusk call for Mr Johnson to set out a detailed plan for Brexit
– Details emerge of Mr Johnson's new Government, with a Cabinet return for Priti Patel and an advisory role for Vote Leave mastermind Dominic Cummings.
– As Mr Johnson wielded the axe, Penny Mordaunt said she was "heading to the backbenches", leaving her post as Defence Secretary, Liam Fox "sadly" left his job as International Trade Secretary and Greg Clark departed his role as Business Secretary.
In his Downing Street address, Mr Johnson was keen to set out a domestic agenda in order to ensure that his term in office is not defined by Brexit.
He confirmed his campaign pledge to put another 20,000 police on the streets, work would start this week on 20 new hospital upgrades and he promised to "fix the crisis in social care once and for all" with a plan to give every older person "the dignity and security they deserve".
Acknowledging the divisions in the country, Mr Johnson said he would answer the pleas of the "forgotten people and the left behind towns", with investment in new transport links and infrastructure.
He also hailed the "awesome foursome" of the four nations of the UK – England, Scotland, Wales and Northern Ireland – "who together are so much more than the sum of their parts".
Mr Johnson faces a difficult task as he attempts to govern with a majority of just two.
The opposition to a no-deal Brexit in the Commons will be bolstered by Mr Hammond, Mr Gauke and Mr Stewart now they are free of Cabinet collective responsibility.
David Lidington, who had been Mrs May's effective deputy prime minister, also stood down as Mrs May resigned and said he had informed Mr Johnson of his decision, saying it was "the right moment to move on" after 20 years on the frontbench in government and opposition.
The influence of the successful Vote Leave campaign in the 2016 referendum is becoming clear as Mr Johnson appoints his government.
Ms Patel was a leading player in the campaign and returns to government after she was forced by Mrs May to resign as international development secretary over unauthorised contacts with Israeli officials.
Mr Cummings clashed with officials and politicians while he was an adviser to Michael Gove in the coalition government, but Mr Johnson clearly believes his forthright style will help steer Brexit through.
The appointment of the abrasive Vote Leave campaign director is controversial given that earlier this year he was found to be in contempt of Parliament for refusing to give evidence to a committee of MPs investigating "fake news".
But uncertainty surrounds the future of Mr Johnson's defeated leadership rival Jeremy Hunt after he reportedly turned down a demotion from Foreign Secretary to defence secretary.