The new prime minister is making a Commons statement, after holding his first Cabinet meeting.
Speaking to MPs, Boris Johnson said his government would throw itself into Brexit negotiations with energy.
He said other arrangements for the Northern Ireland backstop were "perfectly compatible with the Belfast Agreement to which we are, of course, steadfastly committed".
He said Michael Gove would make plans for a no-deal Brexit a "top priority".
He also assured EU citizens they would have "absolute certainty" of their right to live in the UK.
On Thursday, he addressed his cabinet for the first time as prime minister.
Boris Johnson told the cabinet they had "a momentous task ahead", as he repeated his commitment for the UK to leave the EU on 31 October.
On Wednesday, Mr Johnson gave key roles to leading Brexiteers.
The appointments saw Sajid Javid as chancellor, and Dominic Raab and Priti Patel return to government.
Mr Raab was made foreign secretary and Ms Patel is home secretary.
More than half of Theresa May's old cabinet, including leadership rival Jeremy Hunt, quit or were sacked.
BBC political editor Laura Kuenssberg said there were whispers there could be a big offer for European citizens coming out of cabinet this morning.
What has Mr Johnson said?
Speaking to his cabinet Mr Johnson said: "As you all know we have a momentous task ahead of us, at a pivotal moment in our country's history.
"We are now committed, all of us, to leaving the European Union on October 31 or indeed earlier - no ifs, no buts.
"But we are not going to wait until October 31 to get on with a fantastic new agenda for our country, and that means delivering the priorities of the people."
He also told the room it was "wonderful to see this new team assembled here" which respects the "depth and breadth of talent in our extraordinary party".
Mr Johnson also used his first speech as prime minister to reiterate his determination to take the UK out of the EU by the 31 October "no ifs, no buts".
The UK was originally supposed to leave the EU on 29 March but the deadline was moved to 31 October, after MPs rejected Mrs May's withdrawal deal three times.
Who is in Cabinet?
As well as Mr Javid, Mr Raab and Ms Patel, other key appointments included:
- Stephen Barclay: Brexit secretary (retains post)
- Michael Gove: Chancellor of the Duchy of Lancaster, and no-deal Brexit planning
- Ben Wallace: Defence secretary
- Liz Truss: International trade secretary
- Matt Hancock: Health secretary (retains post)
- Gavin Williamson: Education secretary
- Nicky Morgan: Culture secretary
- Andrea Leadsom: Business secretary
- Amber Rudd: Work and pensions secretary (retains post)
- Jacob Rees-Mogg: Leader of the Commons
See the full cabinet here
Following his appointment as Leader of the House, Jacob Rees-Mogg, who led the pro-Brexit Tory European Research Group (ERG), denied there had been a "Leave" takeover of the cabinet.
"Boris is bringing the country together, the party together, through his cabinet appointments," he said.
Jacob Rees-Mogg is making his first appearance in the House of Commons as the leader of the House at Business Questions.
And who lost out?
Mr Johnson's new cabinet saw 17 of Mrs May's former senior ministers being axed or stepping down.
Former Foreign Secretary Jeremy Hunt said he had been offered an alternative role but had turned it down, while leading Brexiteers Penny Mordaunt and Liam Fox were also replaced as defence secretary and international trade secretary respectively.
Both supported Mr Hunt in the Tory leadership contest.
What has been the response?
Labour's shadow chancellor John McDonnell said if Mr Johnson campaigned on the platform of a no-deal Brexit in any forthcoming general election, his own party would "almost certainly be Remain".
However, he told ITV that Labour would still look at any new deal Mr Johnson negotiated with the EU.
"But at the moment I can't see him stitching up a deal that's acceptable either to Labour or to quite a bit of his own side as well - so it looks as though we will then be in a straight situation between a no deal and Remain," he added.
Meanwhile, First Minister Nicola Sturgeon has written to the new prime minister to say it is "essential" Scotland has an alternative option to his Brexit plan - and to indicate she would continue to press for a referendum on Scottish independence.
Independent MP Nick Boles - who resigned the Conservative whip - criticised the cabinet appointments arguing that "the Conservative Party has now been fully taken over top to bottom by the hard right."
"The few elements remaining of the liberal one-nation Conservative style are neutered captives in this cabinet," he said.
Conservative Iain Duncan Smith defended the appointments saying it was important to have ministers in the cabinet who "believe in the project".
When asked about the possibility of an early election he replied "any government worth its salt would prepare for that eventuality".
An early election could happen if the government loses a no-confidence vote. Lib Dem leader Jo Swinson has tabled an early day motion expressing a lack of confidence in the prime minister.
However, such motions are very rarely debated and generally used only as a way of drawing attention to an issue.