Britain and the EU begin tough Brexit talks on Monday, trying to complete one of the most complex negotiations in history in less than two years.
Here are the key issues as Britain's Brexit minister David Davis and the EU's chief negotiator, Michel Barnier of France, meet in Brussels.
- Sequencing -
Britain seems to have tacitly accepted the EU's plan for sequenced talks, which will focus first on the terms of Britain's withdrawal, with negotiations on a future relationship and trade deal coming later.
In her letter triggering the two-year Brexit process on March 29, May had insisted that Britain wanted to discuss them in parallel.
EU officials suggested the apparent climbdown could buy May's unstable government breathing space to actually decide what kind of future relationship it wants.
"The fact they are coming and that they agree to talk about the subjects that we set out, shows that the clash is under control," a senior European official told AFP on condition of anonymity.
The EU has set three priority areas: Britain's exit bill, the rights of EU citizens living in Britain, and Northern Ireland.
But Monday's talks will seek mainly to tie down the timing, with the EU suggesting monthly cycles over the summer.
The aim is to get EU leaders to agree at a summit in October that there is "sufficient progress" on the divorce to move on to future ties.
- The bill -
The issue most likely to torpedo negotiations is Britain's bill for leaving the bloc.
Brussels first mentioned a figure of 60 billion euros (53 billion pounds, $67 billion) but it is now closer to 100 billion, EU sources told AFP.
The EU says Britain must honour its contributions to the bloc's budget, which has already been agreed up to 2020, as well as commitments to development programmes for poorer member states.
But the true figure could be far lower, as the 100 billion does not account for tens of billions that Britain is set to get back in shared assets and rebates.
- Citizens -
The EU wants to secure the rights of more than three million Europeans living in Britain -- and over one million Britons living on the continent.
Currently, Europeans have the right to live, work, study and claim welfare benefits in Britain, as they do anywhere in the 28-nation union.
Brussels insists that those living in Britain now should be able to keep those rights after Brexit.
Aware that this is a priority for the EU, May is reportedly set to make a "generous offer" on the issue early in the talks.
But EU officials have warned her off trying to do this at an EU summit this Thursday, saying it is too soon.
"Many people are afraid that what is presented as a very generous offer will not be seen as very generous, which could really contribute to a negative atmosphere," a European source said.
- Northern Ireland -
The third key issue is the future of the peace process in the British province of Northern Ireland, and the status of the border with the Republic of Ireland.
The EU says it wants to avoid the return of a "hard border" with Ireland that would require passport controls and customs checks, but how that will be possible without Britain staying part of the EU single market or customs union is not clear.
The sensitive issue has been thrown into further doubt by May's efforts to seek a deal with Northern Ireland's ultra-conservative Democratic Unionist Party to stay in power after the British election.
The loyalist DUP has said it will not accept any "special status" for Northern Ireland in the EU after Brexit, which would eliminate one of the leading suggestions for a solution.
- Future -
Barnier wants agreement on the withdrawal, and on a transitional path to a future relationship, by October 2018, so that the European and British parliaments can ratify the deal by Brexit day in March 2019.
What that future relationship will look like remains anyone's guess.
Many in Britain have seen the election result as repudiating May's threats to walk away without a deal.
Speculation has also mounted that she could now seek a softer Brexit, which involves staying in either the EU's single market or customs union.
But EU officials are sceptical that May's position has changed, just as they are doubtful about the feasibility of either option.
An EU-UK trade deal is far from plain sailing, however, with Brussels warning it could take up to seven years after Brexit to agree on one.