The start of the negotiations comes as work continues across the British government to prepare Britain for life outside of the EU.
Britain has less than two years to negotiate the terms of the divorce and the outlines of the future relationship before it is due to leave in late March 2019.
European Union officials suggested the apparent climbdown could buy May's unstable government breathing space to actually decide what kind of future relationship it wants.
For that reason, Brussels wants as a priority to guarantee rights for 3 million European Union citizens in Britain and be paid tens of billions of euros it says London will owe on its departure.
International Trade Minister Liam Fox will travel to Washington on Monday to explore new trade ties - although no formal negotiations are possible until Britain has actually left the bloc.
"To those in the Tory party who call you Remainer Phil, can you say we are definitely leaving the European Union, absolutely on my watch, and we are defiantly leaving the single market on my watch?"
The issue most likely to torpedo negotiations is Britain's bill for leaving the bloc.
No deal would be "very, very bad" for Britain, Hammond said, but worse would be a deal "deliberately structured to punish us, to suck the lifeblood out of our economy". The extent of Britain's exit bill needs to be decided, with the EU27 expected to seek tens of billions of euros they see as London's fair share of programs to which it has committed.
He added: "It's not as if Europe is leaving Britain; Britain wants to leave the EU". However, it seems unlikely he will be able to resist public comment on the Brexit process, which he has described as a "tragedy".
The European Council president will be a key mediator between the other 27 EU leaders for the duration of the negotiations.
David Davis Jeremy Corbyn said Labour favoured a softer Brexit deal. Barnier and Davis will have lunch and then there will be four working groups to discuss the main issues: citizenship, the Northern Ireland border, the financial settlement and other questions.
"We're leaving the EU and because we are leaving the EU, we will be leaving the single market and by the way, we will be leaving the customs union".
After the general election fiasco, May hopes to secure the backing of the ultra-conservative Democratic Unionist Party, which would add the DUP's 10 seats to the Conservatives' 317 in the 650-member House of Commons.
The European Commission were desperate to keep it off the table because it would involve discussions on trade and customs.
What that future relationship will look like remains anyone's guess.
Donald Tusk, the European Council president, has warned there must be "real guarantees" for EU nationals to continue to live, work and study in the United Kingdom and that a "serious British response" was needed if they were to make progress.
But EU officials are sceptical that May's position has changed, just as they are doubtful about the feasibility of either option.
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I didn't feel like today I was playing slower than I was in the other matches. 'I think he's probably a bit less confident. The 30-year-old has won eight of their 10 previous meetings, most recently at the ATP World Tour Finals last November.