Speaking in Brussels, Brexit Minister David Davis said London wanted a “new, deep and special partnership” in the interest of Britons and all Europeans.
“There is more that unites us than divides us,” he said, adding that Britain was looking for a “positive and constructive tone” in the talks.
“So while there will undoubtedly be challenging times ahead of us in the negotiations we will do all that we can to ensure we deliver a deal that works in the best interests of all of our citizens,” he added.
The EU’s chief negotiator, Michel Barnier, said he hoped they could agree a format and timetable on Monday.
The former French minister said his priority was to clear up the uncertainties that the Brexit vote had created.
The talks – which have started almost exactly a year after the 23 June referendum – mark a significant moment in Britain’s complex task of leaving the bloc after more than 40 years of integration.
German Chancellor Angela Merkel promised the EU would listen carefully to Britain’s wishes, saying it was in the interest of both sides to reach a good deal.
Earlier, Boris Johnson said the talks would lead to a “happy resolution” for both sides.
“The most important thing now is for us to look to the horizon… think about the future, and think about the new partnership, the deep and special partnership that we want to build with our friends,” the Foreign Secretary said as he went into a separate meeting of EU foreign ministers in Luxembourg.
Mr Davis and his team are holding meetings with their opposite numbers at the European Commission, which is leading negotiations on behalf of the EU member states.
Speaking ahead of the talks, Mr Davis said the talks would “shape the future of the European Union and the United Kingdom, and the lives of our citizens.”
He said the UK wanted a “deal like no other in history”.
The British team includes the permanent secretary at the Department for Exiting the European Union (DExEU) Olly Robbins; Phillip Rycroft, the department’s second permanent secretary; and Simon Case, the newly appointed director-general of the UK-EU Partnership. Mr Case was principle private secretary to the Prime Minister before talking up this role.
Other officials around the table include Glyn Williams, director-general at the Home Office, who will bring his expertise on immigration issues, and Catherine Webb, a former treasury official, who is director of Market Access at DExEU.
This first meeting will last just a day with a joint news conference expected by both Mr Davis and Mr Barnier when they emerge.
Sky News understands that the UK team has spent the weekend finalising their negotiating strategy which has been called into question following the unexpected election result.
Theresa May had called for voters in the election to choose her “strong and stable leadership” to ensure that the UK had a strong hand in the Brexit negotiations. However, the shock result and loss of a Conservative majority has reopened the debate over what sort of Brexit the UK wants.
Mr Barnier spent the weekend mountain hiking near his home in the French Alps. In a tweet on Sunday afternoon, he said: “Back this weekend in my countryside, Savoie, to draw the strength and energy that the long hike requires…”
Ever since the UK invoked Article 50 in March, triggering its formal intention to leave the EU, the European side has said it was ready to begin negotiations. However, they were delayed to allow for the UK General Election.
Three key issues will dominate the first phase of the talks.
They include the status and rights of EU citizens living in the UK and British citizens living in the EU, the financial commitments the EU expects Britain to pay as it leaves – the so-called “exit bill” – and the question over the border between Northern Ireland and the Republic of Ireland.
When Britain leaves the EU, the Irish border will become the EU’s only land frontier with the UK.
If the UK opts for a Brexit in which it leaves the single market and the customs union, the Irish border would become a closed border unless negotiators can agree on creative solutions.
Once agreement has been made in this phase of the talks, the second phase discussing the future trade relationship can begin.
The EU side has said that this will only happen when “sufficient progress” has been made on phase one and that they will determine the level of progress.
On Sunday, the International Trade Secretary, Liam Fox, flew to Washington DC on his first trade trip since the election.
Dr Fox is meeting a congressional delegation on Capitol Hill in an attempt to pave the way for a UK/US free trade agreement. The UK is unable to sign such deals while it remains a member of the EU but can explore options and develop ties.