At a press conference in Brusels tonight the two negotiators struck a friendly tone as they shared a platform at the end of the first day of the talks.
"We must first tackle the uncertainties caused by Brexit", centering on citizens living on each other's territory, border arrangements between Ireland and the United Kingdom and the amount that Britain stands to pay to get out of its previous European Union commitments, Barnier said.
But many in Brussels fear that London has no real strategy, with May under pressure at home and 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.
Davis said both sides were "off to a promising start" even though the challenges ahead were daunting.
Officials have made clear the Government still wants to negotiate its future trade relationship with the European Union alongside talks on Brexit terms.
The Brexit Secretary said he had been "encouraged by the constructive approach both sides have taken" and revealed he and Michel Barnier have already laid down a detailed framework for the talks.
"Today we agreed on dates, on organisation and on priorities for the negotiation".
The first part of the negotiations on an exit agreement, widely described as the "divorce talks", are due to focus on three elements - the continuing rights of citizens in each other's jurisdictions after Brexit, the financial liabilities of the United Kingdom, and Ireland's border.
According to Barner, opening Brexit negotiations would help tackle the uncertainty caused by Brexit.
And, turning Ms May's "no deal is better than a bad deal" threat against her, Mr Barnier added: "A fair deal is possible and far better than no deal".
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.
About 900,000 British citizens are living in other European Union countries, according to research by the UK's Office for National Statistics, with Spain home to the largest number at 308,805. They should finally tell us what the aim is. "We keep hearing only what they don't want, but we don't have any picture of what future relations will look like".
Austrian Foreign Minister Sebastian Kurz said: "If we don't succeed both sides will lose".
"Johnson urged Europeans to look at the more distant future".
The vote came as a profound shock to Brussels against a backdrop of rising anti-EU sentiment, with many - including now U.S. President Donald Trump - predicting the bloc's eventual break-up. "And we want to try to solve the hard things as well as possible".