Video report by ITV News Political Correspondent Romilly Weeks
Boris Johnson said "the buck stops here" as he promised to deliver Brexit by October 31 during his first speech on the steps of Downing Street.
The Conservative leader reiterated his pledge to take Britain out the EU by Halloween, "no ifs, no buts", during a wide-ranging speech on Wednesday afternoon.
Within hours of arriving back at Downing Street, Mr Johnson started shaping his new Cabinet.
'The buck stops here' new PM promises
In his first speech, an energised Mr Johnson told the gathering media: "I will take personal responsibility for the change I want to see.
"Never mind the backstop, the buck stops here."
He added that the "doubters, doomsters and gloomsters" who "bet against Britain" will be proven wrong.
"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."
Shortly after his visit to Buckingham Palace where the Queen gave him permission to form a new government, Mr Johnson made a number of policy pledges which include:
Delivering Brexit by October 31
Recruiting an additional 20,000 police officers
Start work on 20 new hospital upgrades
Fix the "crisis" in social care
"Level up" funding for primary and secondary schools
Scrape anti-genetic modification rules
Start work on British satellites
Create free ports to create "thousands of high-skilled jobs in left-behind areas"
Start negotiating free trade deals
As Mr Johnson finished his speech, he posed for pictures outside Number 10 and was given round of applause as he made his way past staff and into his new residence.
Carrie Symonds, Mr Johnson's partner, stood alongside members of his staff as they waited for his arrival following his visit to the Palace.
Earlier on Wednesday afternoon, Mrs May met with the Queen after making her final speech as prime minister outside Number 10, where she was joined by her husband Philip.
As Boris Johnson's convoy headed to the Palace to formally ask the Queen for permission to form a government, Greenpeace protesters managed to block the Mall by forming a human chain.
As the Metropolitan Police rounded up the climate change activists, the new Conservative leader was able to make his way through the Palace gates, where he met with the Queen.
Theresa May's final bow
On the steps of Downing Street, Mrs May opened her speech by congratulating Mr Johnson on his leadership win and wished him well as he attempts to secure a Brexit deal in less than 100 days.
"I repeat my warm congratulations to Boris on winning the Conservative leadership election," she said.
Watch Theresa May's full resignation speech outside Number 10
"I wish him and the government he will lead every good fortune in the months and years ahead."
She also thanked those who helped her during her premiership, and said the "immediate priority" was to deliver Brexit for benefit of the nation and to move the UK away from the "current impasse".
Watch ITV News' coverage of the day's events
Mrs May said: "The immediate priority being to complete our exit from the European Union in a way that works for the whole United Kingdom."
During Mrs May's farewell statement as prime minister a shout of "no Brexit" from a heckler could be heard coming from outside the gates of Downing Street to which she replied: "I think the answer to that is - I think not."
Hours earlier, Mrs May appeared emotional as she left the House of Commons to a standing ovation following her last appearance as leader for Prime Minister's Questions.
Chancellor Philip Hammond was among a number of high-profile Cabinet members to resign shortly after Mrs May's Commons bow.
During Mrs May's final tussle at the dispatch box with Jeremy Corbyn, she said the Labour leader had "broken promise after promise" during his general election campaign.
"As a party leader who has accepted when her time was up, perhaps the time is now for him to do the same," Mrs May added.
Who could be in Mr Johnson's government?
While Mrs May and Mr Corbyn traded barbs for the final time, Mr Johnson begun to assemble his top team to help him break the Brexit impasse.
Mr Hammond and Mr Gauke have already warned they are prepared to join other pro-EU Tories in seeking to block a no-deal Brexit.
The new prime minister will outline his new Cabinet on Wednesday evening.