EU's chief negotiator Michel Barnier said he hoped they could agree on a format and timetable on Monday.
United Kingdom negotiator David Davis and the EU's Barnier have one key issue over the first weeks of talks: building trust after months of haggling over leaks and figures over the final bill that Britain would have to pay for leaving.
"We've laid solid foundations for future discussions and an ambitious but achievable timetable", said David Davis, Britain's Brexit minister, adding he was "encouraged" by the first talks.
"To that end we are starting this negotiation in a positive and constructive tone, determined to build a strong and special partnership between ourselves and our European allies and friends for the future", Davis said.
Barnier says the two sides must reach agreement on a deal by October 2018, to give the European and British parliaments time to ratify the deal by Brexit Day in March 2019.
Davis and Barnier have one key issue over the first weeks of talks: building trust after months of haggling over leaks and figures over the final bill that Britain would have to pay for leaving.
"I think the whole process will lead to a happy resolution which can be done with honour and profit to both sides", Johnson said as he went into a separate meeting of EU Foreign Ministers in Luxembourg.
Though less visibly upbeat than veteran Brexit campaigner Davis, Barnier insisted the two sides would work together for a "fair deal" that would not "punish" Britain.
Yet many in Brussels fear that London has no real strategy, with May under pressure at home, still trying to close a deal with a conservative Northern Ireland party to stay in power, and facing criticism for her handling of the aftermath of a devastating tower block fire.More news: Five missing after Grenfell Tower fire found 'safe and well'
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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.
The EU has made its opening position clear.
So far she has balanced the two by having ministers from both wings in her top team or cabinet, and by carefully controlling any message on her approach to Brexit, fearful she could give away her hand to European Union negotiators.
Privately, the British team concedes it will largely have to acquiesce to European Union demands for trade to be left off the agenda for now, although it stresses that no final deal can be reached before all aspects of the future relationship are also considered.
The start of the negotiations comes as work continues across the British government to prepare Britain for life outside of the EU.
"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".
He said it would also have to accept the jurisdiction of the European Court of Justice, "or at least a joint court that is staffed by Europeans and Britons" and in principle follows the ECJ's rulings.
Davis and Barnier, who chatted in English over lunch but used interpreters for some of their more serious talks, made use of their shared love of hill-walking to lighten the mood.