Image source, EPA

Michel Barnier says it is not a matter of the EU making concessions or looking to punish the UK for its decision to leave the EU. This fits in with the general thrust of the day, and the two sides seeking to stress the message that the deal being sought was the best one for both sides.

But, added Mr Barnier, the UK's decided to leave the EU and there will be consequences - "do not underestimate the consequences" - financial, social etc. He then says he will seek to try and take the emotion out of the process and try to minimise the consequences.

It is not about "punishment or revenge". But his warning of consequences makes for what feels like a less fraternal end to the news conference.

Michel Barnier says transparency is of great importance, particularly as Brexit is of huge interest to all 27 members.

"Everywhere I find great awareness of the responsibilities and issues at stake," he says.

David Davis agrees, saying he and the Prime Minister are of the same view and want to see the public informed - by proactive publication of what we are doing, not by leaks from cliques or erroneous briefings.

A journalist asks Mr Barnier "how confident" he is that he will still be dealing with Mr Davis in six months time given the fragility of the Tory government and its reliance on the Democratic Unionists for support.

"How flattering", the Brexit secretary jokes in response. He goes on to say that he hopes an agreement on residency rights can be struck reasonably quickly but that a settlement on issues relating to Ireland will take much longer.

However, he stresses that the UK's relationship with the DUP will not affect progress towards the latter in any what whatsoever.

As for Mr Barnier, he says he has known his counterpart for 20 years since they were EU British and French ministers respectively and that they got on well, adding that the UK is not the only country whose politics he follows closely.

Any Brexit changes to the UK's wildlife laws will increase - not reduce - environmental protection, Michael Gove has pledged.

In response to a question from the BBC's Europe editor Katya Adler, David Davis says nothing has changed in terms of the UK's approach to the single market and customs union from its pre-election blueprint.

In her Lancaster House speech earlier this year, Mrs May indicated that the UK would leave the single market and would seek separate but equivalent customs arrangements.

Mr Barnier echoes this, saying there are "different forms of co-operation" between EU members and third-party countries, adding that the UK "knows what the rules are for those kind of models".

ITV's James Mates asks about the sequencing of talks, noting how they seem to mirror the EU's proposals and says that they suggest a lack of leverage on the UK's part.

Mr Davis says what is important is "not how the talks start, but how they end".

He says that the UK wants talks on a future partnership with the EU to run in conjunction with issues relating to separation, adding that everything announced today is in keeping with what the UK set out in its letter activating Article 50 and the UK's position has not changed.

Other details coming out of a statement on the terms agreed for the talks show that the "principals" - those with overall responsibility for managing the negotiating process - "may decide to establish additional working groups, subgroups or organise breakout sessions".

The document also says each round of negotiations "should comprise public officials of both sides only".

Texts for discussion will have to be shared at least one week ahead fo talks.

We are on to journalist questions. Mr Davis is asked about the Irish border. He says that the issue took up more of the discussion than any other and that is why the two sides' top officials were tasked with overseeing it.

Mr Barnier rejects suggestions that the issue has been "set aside", saying it is highly "sensitive" and the EU is highly conscious of the political situation in Northern ireland and the change of political leadership in the Irish Republic.

All the commitments entered into in the Good Friday Agreement must be honoured and the Common Travel Area protected, he adds.

"We have an awful lot of work to do...so we come up with imaginative and concrete solutions," he adds.

A bit more of the detail coming through about the terms of the Brexit talks...

The negotiating rounds will "consist of plenary sessions and negotiating group meetings".

"Initial negotiating groups" have been set up focusing on citizens' rights, financial settlement and "other separation issues".

The statement goes on: "In addition, a dialogue on Ireland / Northern Ireland has been launched under the authority of the Coordinators".

The UK's Brexit Secretary says this is the first step of a long journey but the two sides have made a "promising start" - adding that substantive progress now needs to be made.

He says there is a long road ahead and then dusts off what he says is a quote of his own from Winston Churchill - a "pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty" - to prove he is a determined optimist.

It is David Davis's turn now. He starts by stressing how positive the talks have been. The timetable set out by the EU is "eminently achievable".

His goal is a strong and special partnership with the EU going forward.

On the key issue of the rights of EU citizens, he says the UK wants an agreement quickly and will set out "its offer" next Monday.

Details released by the EU show that the next stage of talks on Brexit will take place in the week beginning July 17.

The third round will be in the week beginning August 28; the fourth from September 18, while the fifth are set for the week beginning October 9.

Basically there's going to be one week of talks each month during the Brexit negotiating process.