Chief U.K. negotiator David Davis is meeting his European Union counterpart, former French foreign minister Michel Barnier, to grapple with a complex set of questions about the future of trade and migration, how much the country must pay to settle its bill with the bloc, and the rights of millions of citizens who have settled in Britain or Europe.
European Union negotiator Michel Barnier said the negotiations which should lead to a breakup by March 2019 "must first tackle the uncertainties caused by Brexit - first for citizens, but also for the beneficiaries of the European Union policies and for the impact on borders, in particular Ireland".
Brexit talks are officially underway.
Brexit negotiations which could define the UK's political and economic future have begun, with David Davis calling for a "new deep and special partnership" between Britain and Brussels.
The UK's exit from the EU will probably only have a small negative impact on travel within the Union: 40% of UK citizens travel to the EU once a year, and 27% have never been to another EU country. "We must lift the uncertainty caused by Brexit", said Barnier.
Theresa May has said she is "getting on with the job" amid continued questions over her future as Prime Minister.
"They should be agreed alongside each other, this is completely consistent with the Council's guidelines which state nothing is agreed until everything is agreed".
"Nothing is agreed until everything is agreed", he said, using a line also expressed by the EU.
Mr Davis said there was more time spent on Ireland in Monday's talks than any other subject.
Barnier and Davis will have a week of negotiations each month, while the other three weeks will be used for preparations and working out the details.
Barnier noted that the parties to the talks aimed at agreeing on the key challenges of the UK withdrawal from the United Kingdom as soon as possible, including the mutual citizens' rights, financial settlement and the issues of borders, particularly in Ireland.
A detailed paper will then be published on Monday, Mr Davis confirmed.
Speaking after the referendum vote, Mr Barnier said that shouldn't be "prisoner to the British question" during Brexit negotiations.
Both negotiators insisted that the issue was sensitive and would be dealt with at the "highest level".
German Chancellor Angela Merkel says she hopes for a "good agreement" after Brexit talks in which 27 European Union countries will listen carefully to what Britain wants but also defend their own interests.
The Brexit Secretary and EU's chief negotiator appeared together on stage as they face off in the opening round of bruising talks over the UK's exit.
Davis and Barnier will hold a press conference later on Monday.
With a further million British expatriates in the EU, May too wants a deal on citizens' rights, though the two sides are some way apart. We discussed two aspects of it.
"So while there will undoubtedly be challenging times ahead of us in the negotiations, we will do all that we can to ensure we deliver a deal that works in the best interests of all citizens".
"Because membership of the single market requires the four freedoms [of people, goods, services and capital] to be abided, and [because] we need to bring back to Britain control of our laws, control of our borders, we'll be leaving the single market", he said.
While Davis has conceded on trade and divorce bill talks being carried out in parallel from the start, the issue of Ireland's border with the United Kingdom is playing out how he predicted.
Barnier, in turn, said that he would seek "no concessions" because there was none to be found.
"It can mean less scope for a messy fallout between the United Kingdom and the European Union over the divorce settlement".
European Union negotiators told the Times that they fear without some early compromise in the talks, a newly weakened May might not survive the duration of Brexit talks this summer, but believe the Prime Minister might use a private Brussels dinner on Thursday to ease off on some of her previous demands. Mr Barnier quoted one of the EU's founding fathers, Jean Monnet, saying in French: "Ce qui est important, ce n'est, ni d'être optimiste, ni pessimiste, mais d'être détermine". I am determined." Davis channeled Winston Churchill: "The pessimist see difficulty in every opportunity, the optimist sees opportunity in every difficulty.