Summary
May speaks about Brexit at Davos
Says UK to lead world on free trade
Audience is mainly business leaders
IMF's Lagarde warns UK of Brexit pain
UK to trigger EU exit talks by April
Live Reporting
Alex Hunt, Jackie Storer and Emma Griffiths
Watch: EU official says UK customs plans 'ambiguous'published at 16:44 Greenwich Mean Time 17 January 2017
Ken Clarke 'none the wiser' over trade planspublished at 16:44 Greenwich Mean Time 17 January 2017
16:44 GMT 17 January 2017BBC News Channel
Former Chancellor Ken Clarke, one of the most pro-European voices on the Conservative benches, welcomes the "grown-up" tone of Theresa May's speech but says he has been left "none the wiser" as to what the government's plans are to maintain free trade within the internationally accepted framework of enforceable rules.
By seeking a bespoke customs agreement and a new free trade deal outside the European Court of Justice, Mr Clarke says the UK could end up with what it already has but, being outside the EU, find itself unable to influence the direction of policy or contest changes to regulations.
Quote MessageThey were particularly confusing about what we are going to do about access to the single market and membership of the customs union. I listened to David Davis as well as Theresa May and I am none the wiser - and I don't think either of them are either."
Mr Clarke said free movement of people was a vital component of free trade and British businesses could be left worse off if a new immigration system of work permits meant they couldn't recruit who they wanted.
Quote MessageWe are responsible for the single market - it is a Thatcher government idea. We have always had freedom of movement and we would have objected strongly if the French had ever argued that (for work permits). If you are going to trade, you have to have movement of people and then you have to have binding rules...You don't suddenly say that the British will decide what the rules are at any certain time and only our courts will decide any disputes."
How could customs union deal work?published at 16:42 Greenwich Mean Time 17 January 2017
16:42 GMT 17 January 2017Theresa May says she wants an agreement with the customs union but not full membership.
Read MoreMP: Best speech on Europe, everpublished at 16:31 Greenwich Mean Time 17 January 2017
16:31 GMT 17 January 2017Conservative MP Peter Bone says Mrs May's speech was the "best by a British prime minister on Europe - ever" - adding that he is "over the moon" about what he heard.
The EU would be "mad" not to grant the UK a free trade deal, given its 60bn euros trade surplus, he tells the BBC News Channel.
Guy Verhofstadt, the European Parliament's chief Brexit negotiator, says the UK cannot be allowed a better deal than it currently enjoys within the EU to discourage other countries from considering their future.
But Mr Bone says if the EU ignores public opinion and tries to impose a punitive deal on the UK, this will be the "death knell" for the organisation.
The UK's Brexit plans: What we now knowpublished at 16:30 Greenwich Mean Time 17 January 2017
16:30 GMT 17 January 2017Theresa May has revealed more about what Brexit might mean for the UK - here's what we know now.
Read MoreSturgeon: No mandate for single market exitpublished at 16:27 Greenwich Mean Time 17 January 2017
16:27 GMT 17 January 2017Nicola Sturgeon says a second Scottish independence referendum is now closer after Theresa May laid out her Brexit objectives.
The Scottish first minister accused Theresa May of driving the UK off a "hard Brexit cliff edge" and said leaving the single market and the customs union would be economically damaging for the whole country.
She also insisted that the prime minister had no mandate to take the UK out of the single market and argued that a choice was emerging about whether Scotland wanted to be steamrollered down a path it didn't want to take, or to take control of its own future.
Key quote: May warns EU of risks of not dealing better with diversity of memberspublished at 15:49 Greenwich Mean Time 17 January 2017
15:49 GMT 17 January 2017Quote MessageBritain is not the only member state where there is a strong attachment to accountable and democratic government, such a strong internationalist mindset, or a belief that diversity within Europe should be celebrated. And so I believe there is a lesson in Brexit not just for Britain but, if it wants to succeed, for the EU itself. Because our continent's great strength has always been its diversity. And there are two ways of dealing with different interests. You can respond by trying to hold things together by force, tightening a vice-like grip that ends up crushing into tiny pieces the very things you want to protect. Or you can respect difference, cherish it even, and reform the EU so that it deals better with the wonderful diversity of its member states."
Listen: Final deal will depend on what the UK can offerpublished at 15:45 Greenwich Mean Time 17 January 2017
15:45 GMT 17 January 2017BBC radio's Newshour
British Prime Minister Theresa May has announced that the UK will seek a trade deal with the European Union, rather than remain in the single market. So how easy will it be for British negotiators to access the free trade bloc?
Dr Norbert Röttgen is from Germany's governing CDU party and is chairman of the Bundestag's Committee on Foreign Affairs - he says that the final deal will depend on how much Britain can offer the EU.
Key quote: Theresa May on a customs unionpublished at 15:42 Greenwich Mean Time 17 January 2017
15:42 GMT 17 January 2017Quote MessageI know my emphasis on striking trade agreements with countries outside Europe has led to questions about whether Britain seeks to remain a member of the EU's Customs Union. And it is true that full Customs Union membership prevents us from negotiating our own comprehensive trade deals. Now, I want Britain to be able to negotiate its own trade agreements. But I also want tariff-free trade with Europe and cross-border trade there to be as frictionless as possible. That means I do not want Britain to be part of the Common Commercial Policy and I do not want us to be bound by the Common External Tariff. These are the elements of the Customs Union that prevent us from striking our own comprehensive trade agreements with other countries. But I do want us to have a customs agreement with the EU. Whether that means we must reach a completely new customs agreement, become an associate member of the Customs Union in some way, or remain a signatory to some elements of it, I hold no preconceived position. I have an open mind on how we do it."
EU statement begins in the Lordspublished at 15:41 Greenwich Mean Time 17 January 2017
15:41 GMT 17 January 2017House of Lords
ParliamentBrexit Minister Lord Bridges of Headley is repeating to the Lords the statement made earlier by David Davis, in the Commons.
Labour's spokeswoman Baroness Hayter welcomes the objective of ending up with a fairer Britain, but adds that it would have been "odd" if an unfair Britain had been the government's goal.
She fears a "sting in tail" in the government's belief that "no deal is better than a bad deal". That sounds like "lower taxes and fewer public services", she says.
Scotland and the EU: SNP MPs ask about future dealpublished at 15:32 Greenwich Mean Time 17 January 2017
15:32 GMT 17 January 2017Brexit statement
House of Commons
ParliamentThe SNP's Neil Gray says that the UK government's position is incompatible with the Scottish government's Scotland's Place in Europe document, external.
He asks how the UK government proposes to honour the promise to take Scotland's proposals seriously and to explore all options to ensure continued Scottish membership of the single market.
SNP MP Joanna Cherry says that as EU officials have been prepared to contemplate a special deal for the city of London, there is scope for a differentiated deal that the Scottish government seeks.
On the issue of the Scottish government document, David Davis says it contains questions about devolution, employment and immigration - all of which the government will treat seriously, "as we always have".
Mr Davis also says that there have been no special deals with any sector and that the aim of the British government is that the whole economy succeeds and not just one part of it, and that includes Scotland too, he says.
Brexit means...published at 15:28 Greenwich Mean Time 17 January 2017
15:28 GMT 17 January 2017Brexit means Brexit, but what does the prime minister's Brexit speech mean for Wales?
Read MoreUKIP’s Carswell: May speech could ‘win over remainers’published at 15:28 Greenwich Mean Time 17 January 2017
15:28 GMT 17 January 2017BBC Radio 5 live
UKIP’s Douglas Carswell has told BBC Radio 5 live he was “really, really pleased” with Prime Minister Theresa May’s speech and it could “win over some of the remainers too”.
The MP for Clacton said the Prime Minister’s speech was a quite “liberal, pragmatic, sensible” approach to Brexit and he could not find “anything wrong with it”.
Asked by presenter Nihal Arthanayake if he would consider rejoining the Conservative Party after hearing Mrs May’s speech, Mr Carswell said: “If my party with one MP in the House of Commons can get the system to change its mind and see sense on the Europe question, think of all the other things that we could change.”
EU negotiator Barnier 'ready' for actionpublished at 15:27 Greenwich Mean Time 17 January 2017
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End of twitter post 2Johnson praises May's 'exciting vision' for UKpublished at 15:27 Greenwich Mean Time 17 January 2017
15:27 GMT 17 January 2017Foreign Secretary Boris Johnson has given his reaction to the prime minister's speech in a video posted to his Twitter account, external, in which he says:
"I've just been listening to Theresa May's fantastic speech about global Britain and about how we're going to take back control of our borders and loads of money that we currently send to the EU, but also of course come out of the EU's legal system, the single market, but not leave Europe.
"We're still going to be committed to the safety, the security of our continent, still determined, with our friends, to build a new European partnership outside the European Union that simultaneously enables us to build a future as a great free-trading nation again.
"And it's a very, very exciting vision for this country."
Villiers: Article 50 is 'irreversible'published at 15:27 Greenwich Mean Time 17 January 2017
15:27 GMT 17 January 2017BBC News Channel
Former Northern Ireland Secretary Theresa Villiers says that while the UK has a number of options for determining trade with the EU after Brexit it is clear that remaining in a full customs union is "off the cards".
She tells the BBC that it remains to be seen whether what emerges is a bespoke customs agreement or something else but the rest of the EU would find it hard to reject a "reasonable and sensible" deal based on reciprocity.
Asked what will happen if Parliament rejects the overall Brexit deal negotiated by Mrs May, she says her understanding is that the Article 50 process is "irreversible" and that the UK would leave the EU anyway.
Listen: What is a customs union?published at 15:18 Greenwich Mean Time 17 January 2017
15:18 GMT 17 January 2017The World at One
BBC Radio 4Tory MP - Brexit offers a 'wonderful opportunity'published at 15:16 Greenwich Mean Time 17 January 2017
15:16 GMT 17 January 2017Brexit statement
House of Commons
ParliamentConservative MP Gareth Johnson says that around 40% of UK exports go to the EU, and says that this is "part of the problem", as only 7% of the world's population lives in the European Union.
The referendum decision to leave the EU therefore offers a "wonderful opportunity" to be a more global and international nation, he says.
David Davis responds that growth rates in global markets beyond Europe are much higher than in the European Union, which is one of the opportunities that arises from leaving the EU.
Questions on accommodating Remain voters in Northern Irelandpublished at 15:16 Greenwich Mean Time 17 January 2017
15:16 GMT 17 January 2017Brexit statement
House of Commons
ParliamentThe SDLP's Alasdair McDonnell says that he hopes the government recognise that thousands of people in Northern Ireland will not be leaving the EU willingly as they recognise the "very significant benefits" that have come from EU membership.
"We hold EU passports and we intend to retain them," he says, asking "what will be done to accommodate us?"
Brexit Secretary David Davis tells MPs that since the beginning of the process the government has put the preservation and stability of Northern Ireland "pretty much at the top of the tree" - with the maintaining of an open border and other issues having priority.
Remove students from immigration figures - Labour MPpublished at 15:16 Greenwich Mean Time 17 January 2017
15:16 GMT 17 January 2017Brexit statement
House of Commons
ParliamentLabour Leave campaigner Gisela Stuart says that if MPs are seeking a subject to unite both sides of the Brexit divide, then she suggests that Mr Davis should talk to the Home Office, and ministers who deal with universities, to remove the number of international students from the head count of immigration figures.
Brexit Secretary David Davis says that this is more properly a question for the Home Office, but says to Ms Stuart that she can be sure that the operation of immigration policy after the UK leaves the EU will be in the national interest.
"That includes the interests of our incredibly powerful and effective university sector," he says.