Mr Davis and Mr Barnier, both keen hikers, exchanged mountaineering-related gifts to mark the start of talks, with the EU's chief negotiator joking about the "steep and rocky path" ahead.
Chancellor of the Exchequer Philip Hammond said on Friday that protecting jobs and the British economy would be priorities for his country.
GettyWhen are Brexit talks?
It certainly is. Mr Davis has said that while the United Kingdom is pulling out of the single market and the customs union and ending the free movement of labour, he wants an agreement that replicates as far as possible Britain's existing trade arrangements with the EU.
A statement issued by Davis' department Sunday in London said "Despite European leaders' attempts to leave open the possibility of Britain remaining in the UK, Davis will make it clear that he is determined to achieve a Brexit deal that works for the whole of the UK".More news: Rahul Dravid Set To Continue As Under 19 & India A Coach
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Both EU negotiator Michel Barnier and his British counterpart David Davis said after the first negotiating session they were confident of quick progress but said major challenges lay ahead to meet the deadline of March 2019 for Britain to officially leave the bloc.
It's been almost a year since British voters approved the referendum known as Brexit to leave the European Union and almost three months since Prime Minister Theresa May formally launched the separation process.
He said that the first phase of Brexit talks will include the rights of citizens, the Brexit bill and the issue of the border in Ireland.
The UK kicked off talks in Brussels with both sides agreeing a schedule for dialogue as they attempt to pave the way to Brexit. "Our aim is to have one week of negotiations every month", said Barnier that further described the break-down of talks to into three groups that are to deal with: Citizens rights, Single financial settlement and Other separation issues. Both sides will put top advisers to work immediately on a border agreement between Ireland and the United Kingdom, aiming to make sure the Irish peace agreement and the common travel area should as unaffected by Britain's European Union departure as possible. But a government spokesman in London said, "We believe that the withdrawal process can not be concluded without the future relationship also being taken into account".
"The pessimist see difficulty in every opportunity, the optimist sees opportunity in every difficulty".
The UK has begun discussions with the European Union on Brexit, despite uncertainty as to who will form the government that will negotiate the exit deal.
"So now that the negotiations have started, we are determined to get on with the job and deliver that certainty as soon as possible". However, not much clarity was provided immediately after the first official meeting of the EU-UK sides.