"We 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", Barnier told reporters at the start of the talks.
"We will build the broadest possible consensus for our Brexit plans and that means giving Parliament the maximum amount of time to scrutinise these bills by holding a two-year session of Parliament", she said.
Here are the key questions surrounding the Brexit negotiations.
After seven hours of talks in Brussels, Mr Davis - who had previously promised the "row of the summer" over the timetable for the negotiations - said he was optimistic about the talks.
A memo sent to the EU's Committee of the Regions state: "No suits and ties where possible", "You are advised not to drink alcohol" and "Switch off the lights".More news: Steinberg: Next Cosby Trial Could End The Same Way
Both sides are still in disagreement over how exactly they will progress.
As required by the European Union, negotiations will start with the divorce itself, and when "sufficient progress" has been achieved, they will move onto discussing the future relationship between the bloc and the UK.
"Theresa is leading the government and I think the government needs to get on with the job", he said.
It's vital that the deal we strike allows both the United Kingdom and the European Union to thrive as part of a new deep and special partnership we want with our closest allies and friends.
Negotiations start exactly one year after Britain shocked the world by voting to exit the European Union and almost three months since Prime Minister Theresa May locked the country into a two-year countdown to Brexit in March 2019.
"The future of UK Prime Minister Theresa May continues to be shrouded in uncertainty as whispers continue to grow as to the length of her tenure, ahead of this week's scheduled Queen's Speech".More news: Oracle skipper Spithill tough on, off water in America's Cup
The Mechanical Engineering Industry Association, known by its German initials VDMA, says that that goal of the two-year negotiating process is "damage limitation" because Brexit won't benefit either side.
But on Sunday, he also backed up May's approach, saying Britain would leave the single market and the customs union that sets external tariffs for goods imported into the bloc. He said that the other European Union countries have a united position but the British are "in chaos".
The storm clouds began to gather in the immediate aftermath of the referendum, when investors quickly sent the pound to its lowest level in decades over fears that Britain would lose preferential access to Europe's vast markets.
"Obviously this is the first day of the talks on Brexit and I think the most important thing is we should all start - of course there'll be lots of discussions about the nature of the deal we are going to do - but I think we should also enter on the discussion about money and so forth", he said.
As the negotiations kick off, here's a look at some of the major issues the sides face.
Other members of May's Conservatives have called for a more inclusive approach on Brexit strategy that would include the voices of opposition parties as well as the views of Scotland and Northern Ireland, which both voted to stay in the EU.More news: Trump's legal plan quite simple so far: Fight, fight, fight
British foreign minister Boris Johnson, like Davis a prominent backer of the leave campaign, also sounded an upbeat note.
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