UK's land border with Ireland and EU should be as seamless as possible - Theresa May
* Joint news conference with Theresa May and Taoiseach Leo Varadkar
* Negotiators Davis and Barnier met at 11 am (0900 GMT)
* Main aim of day's talks to agree format and build trust
* May's election debacle leaves British plans unclear
* EU heartened by new president Macron's sweep in France
Britain's Prime Minister Theresa May said that she wanted the border between the United Kingdom and the European Union to be as seamless as possible after Brexit.
The border between Northern Ireland, which is part of the UK, and the Republic of Ireland, an EU member, will become the only land border between the UK and the EU after Britain exits the bloc in March 2019.
Speaking at a joint news conference with May, Taoiseach Leo Varadkar said that the border should be "invisible".
May is currently seeking backing from the DUP, a Northern Irish party, for her minority Conservative government after she lost her parliamentary majority in a June 8 election.
Varadkar told the news conference that both the British and Irish governments needed to be impartial actors in relation to Northern Ireland's stalled power-sharing arrangements between parties that want the province to remain in the UK and parties that want it to become part of the Republic.
The chief Brexit negotiators of the European Union and Britain began talks on Monday morning by stressing the need to quickly tackle uncertainties in the process and underlining their constructive attitude to reach a deal that is good for all.
The European Union's Michel Barnier said he hoped the talks, starting almost a year to the day after a British referendum vote to leave the EU, would establish a timetable for the negotiations.
"We must first tackle the uncertainties caused by Brexit - first for citizens but also for the beneficiaries of the EU policies and for the impact on borders, in particular Ireland," Barnier told reporters at the start of the talks.
"I hope that today we can identify priorities and a timetable that would allow me to report to the European Council later this week that we had a constructive opening of negotiations," he said.
Britain's Brexit minister David Davis said London wanted a deep and special relationship with the EU after the divorce and that he would conduct the talks in a constructive tone.
"There’s more that unites us than divides us, so while there will undoubtedly be challenging times ahead of us in the negotiations, we will do all that we can to ensure that we deliver a deal that works in the best interests of all citizens," Davis said.
"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," he said.
Almost a year to the day since Britons shocked themselves and their neighbours by voting on June 23 to cut loose from their main trading partner, and nearly three months since Prime Minister Theresa May locked them into a two-year countdown to Brexit in March 2019, almost nothing about the future is clear.
Even May's own immediate political survival is in doubt, 10 days after she lost her majority in an election.
Davis, who unlike May has long campaigned to leave the EU, will meet chief EU negotiator Michel Barnier, a former French minister, at the European Commission's Berlaymont headquarters at 11 a.m. (0900 GMT). They are due to give a joint news conference after talks among their teams lasting seven hours.
Officials on both sides play down expectations for what can be achieved in one day. EU diplomats hope this first meeting, and a Brussels summit on Thursday and Friday where May will encounter - but not negotiate with - fellow EU leaders, can improve the atmosphere after some spiky exchanges.
"Now, the hard work begins," Davis said, adding he wanted a deal that worked for both sides.
"These talks will be difficult at points, but we will be approaching them in a constructive way."
Barnier, a keen mountaineer, spent the weekend in his native Alps "to draw the strength and energy needed" ahead of Brexit talks.
Davis's agreement to Monday's agenda led some EU officials to believe that May's government may at last coming around to Brussels' view of how negotiations should be run.
Sounding conciliatory, Britain's Boris Johnson said as he arrived at a meeting with fellow EU foreign ministers in Luxembourg that he looked forward to "a happy revolution" in relations that would be good for Britain and the rest of Europe.
"The most important thing I think now is for us to ... think about the new partnership, the deep and special partnership that we want to build with our friends," said Johnson, who campaigned in last year's referendum to leave the EU.
May's election debacle has revived feuding over Europe among Conservatives that her predecessor David Cameron hoped to end by calling the referendum and leaves EU leaders unclear on her plan for a "global Britain" which most of them regard as pure folly.
While "Brexiteers" have strongly backed May's proposed clean break with the single market and customs union, finance minister Philip Hammond and others have this month echoed calls by businesses for less of a "hard Brexit" and retaining closer customs ties.
With discontent in europhile Scotland and troubled Northern Ireland, which faces a new EU border across the divided island, Brexit poses new threats to the integrity of the United Kingdom.
It will test the ingenuity of thousands of public servants racing against the clock to untangle 44 years of EU membership before Britain is out, 649 days from now, on March 30, 2019. For the officials sitting down on Monday, at least on the EU side, a major worry is Britain crashing out into a limbo, with no deal.
For that reason, Brussels wants as a priority to guarantee rights for 3 million EU citizens in Britain and be paid tens of billions of euros it says London will owe on its departure.
With a further million British expatriates in the EU, May too wants a deal on citizens' rights, though the two sides are some way apart. Agreeing to pay a "Brexit bill" may be more inflammatory.
Brussels is also resisting British demands for immediate talks on a future free trade arrangement. The EU insists that should wait until an outline agreement on divorce terms, ideally by the end of this year. In any case, EU officials say, London no longer seems sure of what trade arrangements it will ask for.
But Union leaders, including German Chancellor Angela Merkel and French President Emmanuel Macron, are also determined not to make concessions to Britain that might encourage others to follow.
When 52pc of British voters opted for Brexit, some feared for the survival of a Union battered by the euro crisis and divided in its response to chaotic immigration. The election of the fervently europhile Macron, and his party's sweep of the French parliament on Sunday, has revived optimism in Brussels.
German Chancellor Angela Merkel said she wanted a "good agreement" for both Britain and the European Union in the Brexit negotiations.
"For me, it is above all about the EU27 proceeding together and listening carefully to Britain's wishes and expectations," Merkel told a news conference after meeting Romanian President Klaus Iohannis.
"And it is also about representing our own interests and progressing towards a good agreement. But it is too early to speculate about the details. I want us to reach a good agreement and this should be in the interests of both sides," she added.