The two-day EU summit is over, bringing to a close a busy week for UK Prime Minister Theresa May.

And this brings an end to our live coverage of the afternoon's press conferences marking the end of the event.

Scroll down to catch up on what's happened and what's been said. Thanks for joining us.

Adam Fleming
Brussels reporter

Theresa May’s mission to Brussels has not been a failure.

She has a written statement from her 27 fellow leaders confirming – reconfirming, really – that the Irish backstop is an insurance policy which would only ever be temporary.

If the backstop is activated, then the EU would use their “best endeavours” to negotiate a trade deal which would mean it could be deactivated.

That wording is crucial, say British officials, because it means the UK could refer the EU to the independent arbitration panel established in the Brexit treaty if London felt Brussels was moving too slowly.

The EU dropped a commitment to look for further ways to help the UK, which means there won’t be a formal process to find them.

But it doesn’t mean the search couldn’t happen informally, or in private, or at the last minute.

The problem is that these commitments are unlikely to impress Mrs May’s harshest critics.

And they certainly wouldn’t fit on the side of the bus as reasons to sign up to her Brexit deal.

This afternoon has seen press conferences held by UK Prime Minister Theresa May, European Council president Donald Tusk and European Commission president Jean-Claude Juncker, as well as other EU leaders.

Here's what we learned from their statements, and the reaction that came afterwards:

Theresa May

• Britain will have further talks with the EU in coming days about how to obtain "further assurances" that could persuade MPs to vote in favour of her withdrawal agreement
• There is "work to do" and talks on "clarification" will continue
• It's not possible to reopen the agreement to alter the backstop provisions
• She and Mr Juncker had a "robust" discussion in which she was clear that the UK needed firmer assurances the backstop can't become permanent

Jean-Claude Juncker

• He said when he'd referred to the word "nebulous" - something that sparked a bit of a to-do between him and Mrs May - he was talking about the "overall state of the debate in Britain" and not the UK prime minister himself
• European leaders have the "highest respect" for Mrs May he said, describing her as a friend

Donald Tusk

• The EU has treated Mrs May with more respect than many British MPs
• There is no mandate for any further negotiations

German Chancellor Angela Merkel

• Britain has to take the next steps - the 27 member states have given assurances, and their position is on the table

Labour leader Jeremy Corbyn

• Mrs May's deal is "dead in the water"

DUP leader Arlene Foster

• Mrs May should "stand up" and "not roll over" to EU leaders

EU Commission chief Juncker explained what he meant by his controversial "nebulous" remark in relation to Brexit.

Theresa May challenged him about it, as seen in a video tweeted earlier. Their tete-a-tete caused a huge media stir. Here is his explanation.

The Independent's political sketch writer Tom Peck has a few words to say about that exchange between Theresa May and Jean-Claude Juncker earlier.

You can read the rest of his piece here, external.

Democratic Unionist Party (DUP) leader Arlene Foster has called on Theresa May to "stand up" and not "roll over" to EU leaders during Brexit talks.

She was speaking after Mrs May met with European Union leaders in Brussels.

Mrs May wants legal assurances on the Irish backstop to help the deal get through Parliament, after she delayed a Commons vote in anticipation of defeat.

European Commission president Jean-Claude Juncker said there could be clarifications but no renegotiation.

Read more here.

EU Commission chief Jean-Claude Juncker was filmed enthusiastically ruffling the hair of a woman at the summit today.

BBC colleagues in Brussels managed to identify her as Pernilla Sjölin, deputy head of protocol at the European Commission. So she is part of Mr Juncker's large team.

Here's some more from the Taoiseach, speaking about Theresa May's agreement.

Mr Varadkar said they would begin negotiations on the future as soon as possible once the withdrawal treaty was dealt with.

A German video news agency has shared this video from earlier in the day, showing Jean-Claude Juncker greeting a woman in an unusual way - by ruffling her hair.

It's not clear who the woman is or what prompted the hair ruffle.

On his way in to the final summit press conference the European Commission president was then seen covering a camera lens with his hand - could the two incidents be related?

Image source, Reuters

The Irish premier says he's "very satisfied" with the summit conclusions on Brexit which made clear that the Withdrawal Agreement could not be renegotiated.

Taoiseach Leo Varadkar said:

European Council President Donald Tusk ruled out reopening negotiations on the UK withdrawal deal.

"I have no mandate to organise further negotiations, no mandate to reopen negotiations on the withdrawal agreement," he told the news conference.

But he said: "I’m always at Prime Minister May’s disposal" and "it's my pleasure to be in contact with the prime minister on all relevant issues".

"Yesterday's conclusions are quite clear I think and we have to treat it as a good sign that we are ready to reconfirm our assurances and goodwill and good faith when comes to the so-called backstop."

Mr Juncker is asked about THAT chat with Mrs May again.

He does a little jig on the stage but then says they "weren't dancing".

He then went on to say this:

Mr Tusk says the EU has treated Theresa May "with much greater empathy and respect than some British MPs, for sure".

Mr Juncker adds: "She's a good friend of ours. We have the highest respect for the British prime minister."

Austrian Chancellor Sebastian Kurz - current EU Council president - says that Brexit talks had "overshadowed" all other business at the Brussels summit.

He says: "The last few days have been useful to discuss certain questions with Theresa May and create some more clarity.

"There will be a vote in the British Parliament in January and the deal that was negotiated is a good deal for both sides. We hope that this deal will find support in both parliaments eventually."

Mr Juncker acknowledges there is deep mistrust in the Commons of the EU's position.

He wants to make clear how keen they are to get on with trade talks - which in theory could mean backstop not needed.

He says he won't respond to those attacks from Westminster, but adds: "although I would like to do it".

Following their conversation which was videoed this morning, which appeared to be rather heated, Mr Juncker stresses that Mrs May is his "friend".

Mr Juncker has replied to a question about the future of Brexit.

He says of Mrs May: "This is a woman of great courage, doing her job in the best way possible."

He adds the backstop is not intended to be a "permanent instrument".

Mr Juncker adds that "we have to bring down the temperature" following "attacks" coming from Westminster.

Brexiteers have held a protest on London's Westminster Bridge.

But many have also suggested from a clip they posted on Twitter that they also blocked an ambulance with its lights flashing.