In short press statements, the EU's chief negotiator, Michel Barnier, said he hoped Monday's discussions can "identify priorities and timetable that would allow me to report...that we had a constructive opening of negotiations".
In choreographed talks that saw the two men exchange mountaineering gifts, they agreed to discuss divorce issues before negotiations on a future trade deal can start.
Britain enters the negotiations with U.K. Prime Minister Theresa May facing huge political challenges at home following an election which saw her Conservative party lose its majority in Parliament. In a bid to curb the problems arising from uncontrolled immigration as well as the loss of sovereignty, Britain had voted to end its four decades-long membership of the 28-country bloc, the European Union in the previous year.
Davis, a veteran campaigner against European Union membership, said he sought quick and substantive progress in what is scheduled to be a two-year negotiation before Britain leaves the EU.
Guy Verhofstadt, the EU's Brexit negotiator said "If you want the advantages you of a Single Market and Customs Union, you have to take the obligations", meaning the United Kingdom would have to stay within its confines if it wished to have the benefits.
May's government said it was "confident it can achieve a bold and ambitious deal that will work in the interest of the whole U.K".More news: Confederations Cup: Germany too good for Socceroos in Russian Federation , winning 3-2
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Mr Hammond also drew plaudits for stressing the importance of transitional arrangements to avoid a "cliff edge" Brexit, including the maintenance of "frictionless" customs union border arrangements for an "implementation period" after leaving the bloc.
The Sun, which backed May's centre-right Conservatives in a June 8 election that cost her her parliamentary majority, also noted Davis had gone along with Brussels' plans after saying he would make getting immediate trade talks "the fight of the summer".
The document released after the first day of discussion on "Terms of Reference for the Article 50 TEU negotiations" stated: "The following initial negotiating groups have been established: Citizens' rights; Financial Settlement; Other Separation issues".
In Berlin, German Chancellor Angela Merkel emphasized the unity of the remaining 27 European Union countries, who have been alarmed in recent weeks by May's threats to walk out of the talks.
He said: "It is our priority, it is citizens first".
"We must push for a new phase of globalisation to ensure that it delivers clear benefits for ordinary working people in developed economies".
The chancellor - strengthened since the general election - gave the greatest detail yet about what his approach might mean for our future relationship with the EU.