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 new Tory leader said he would meet the October 31 deadline "no ifs or buts".
Watched by girlfriend Carrie Symonds, 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, Johnson vowed to prove the Brexit doubters wrong.
Johnson – whose progress to his meeting with the Queen was briefly disrupted by climate change protesters – 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 - 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."
Theresa May had earlier used her farewell address in Downing Street to urge 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".
One of May's final acts as prime minister was to receive the resignations of Philip Hammond, David Gauke and Rory Stewart from their Cabinet roles.
Her effective deputy prime minister David Lidington also announced he was standing down from the Government.
Hammond, Gauke and Stewart strongly oppose a no-deal Brexit and say they cannot support Johnson's commitment to take Britain out of the EU by the deadline of October 31 "do or die".
In his resignation letter, Hammond said the new PM should be "free to choose a chancellor who is fully aligned with his policy position".
And in a pointed message to Johnson, he warned that headroom built up in the public finances could only be used for tax cuts and spending boosts if a Brexit deal was secured.
Gauke used his resignation letter to say: "Given Boris's stated policy of leaving the EU by October 31 at all costs, I am not willing to serve in his Government.
"I believe I can most effectively make the case against a no-deal Brexit from the backbenches."
Lidington stood down as May resigned and said he had informed Johnson of his decision, saying it was "the right moment to move on" after 20 years on the frontbench in government and opposition.
He said he would do all he could to help the new government "secure a deal to allow an orderly departure from the EU".
Foreign Secretary Jeremy Hunt later quit the cabinet after losing out to Johnson in the leadership race.
Defence Secretary Penny Mordaunt is another who is returning to the backbenches with Westminster sources saying she was sacked by Johnson.
She tweeted: "I'm heading to the backbenches from where the PM will have my full support, as will my successors at @DefenceHQ & @WomenEqualities.
"Thank you to everyone who's helped me get things done, especially our Armed Forces and civilians in defence for the last 85 days. We achieved much."
International Trade Secretary Liam Fox also said he will be leaving the Government.
He tweeted: "Sadly, I will be leaving the Government. It has been a privilege to have served as Secretary of State for International Trade these past 3 years.
"I am proud to have worked with a tremendously talented team @tradegovuk to boost trade & investment and prepare for Brexit.
"The world-class trade department we have built leaves the UK uniquely well-positioned to forge our new trading relationships beyond Europe and create a truly Global Britain.
"But we must first undertake the momentous task of delivering on the instruction of the British people and leave the European Union. I look forward to supporting @BorisJohnson and the government from the backbenches."
Johnson has been busy in his preparations for government, with a return expected for Eurosceptic Priti Patel and an advisory role for Leave campaign mastermind Dominic Cummings.
Patel is an ardent Brexiteer who was forced by May to resign as international development secretary over unauthorised contacts with Israeli officials.
Cummings clashed with officials and politicians while he was an adviser to Michael Gove in the coalition government, but Johnson clearly believes his forthright style will help steer Brexit through.
The appointment of the abrasive Vote Leave campaign director will be 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".
He is also less than impressed with the calibre of Brexiteer MPs, describing a "narcissist-delusional subset" of the European Research Group (ERG) as a "metastasising tumour" that needed to be "excised".
Johnson will need the support of those same ERG hardliners for his Brexit plan.
Patel has reportedly been lined up for the post of home secretary as allies said Johnson was determined to create a "Cabinet for modern Britain", with a record number of ethnic minority ministers and more women attending in their own right.
It is likely to mean a promotion for the Indian-born employment minister Alok Sharma, who is expected to take his place around the top table.
But uncertainty surrounds the future of Johnson's defeated leadership rival Jeremy Hunt after he reportedly turned down a demotion from Foreign Secretary to defence secretary.
In her final session of Prime Minister's Questions, May used an ill-tempered set of exchanges with Labour leader Jeremy Corbyn to call for him to also quit his post.
After Corbyn accused her of a string of policy failures and U-turns, May told the opposition leader: "Perhaps I could just finish my exchange with him by saying this: As a party leader who has accepted when her time was up, perhaps the time is now for him to do the same?"
