Forming a government
Theresa May will barely have time to unpack her bags before she will face her first test - who she will choose to sit alongside her around the Cabinet table. The most important appointments - chancellor, foreign secretary and home secretary - are expected to be announced on Wednesday with a fuller government reshuffle following over the next couple of days.
Mrs May will be expected to reward allies who supported her in the leadership contest but also maintain a balance between Brexiteers and Remainers in the Cabinet. She will need to find room for some new faces while also maintaining a core of experience.
Mrs May has floated the idea of a new government department to take day-to-day charge of negotiating the UK's exit from the EU - to be headed by someone who campaigned to leave in the referendum. Chris Grayling, a prominent Brexiteer who supported her leadership bid, acting as her campaign manager, has been touted as a possible candidate for this role. Liam Fox, another Leave supporter who got behind Mrs May, after his own leadership bid failed, will also have hopes of a top job.
Brexiteers would also be keen to see Andrea Leadsom rewarded with a prominent role, after giving way to Mrs May in the leadership contest, although the crucial role of chancellor could go to someone with more experience. May ally and Remainer Philip Hammond, currently foreign secretary, is seen by some as the favourite.
Then there is the dilemma of what to do with some of the "big beasts" of the party, such as George Osborne, Michael Gove and Boris Johnson. Will they want to stick around and serve under her given each had leadership ambitions of their own?
Speaking to foreign leaders
Protocol dictates that a new prime minister will get messages of support and congratulations from fellow leaders across the UK and the world. She will only be able to return some of these calls at this stage but the initial conversations she has will be more than mere niceties.
Mrs May will want to get off on the right foot, particularly in her dealings with German Chancellor Angela Merkel and French President Francois Hollande ahead of years of what will be hard-fought negotiations over the UK's divorce from the EU.
And then there will also be the symbolically huge moment of her first conversation with US President Barack Obama.
Although Mr Obama only has seven months left in the White House, Mrs May will want to create an immediate impression on him by emphasising the strength of the "special relationship" and addressing US concerns about the implications of the UK's Brexit vote on the UK's international defence and security commitments.
The UK's exit from the EU will dominate Theresa May's premiership. Despite backing the Remain campaign in the EU referendum, she has said "Brexit means Brexit," and has vowed to make it work.
She has said she will not trigger Article 50 of the Lisbon Treaty, which would formally take Britain out of the EU after up to two years of negotiations, before the end of 2016.
She is thought to want informal talks with other EU leaders first.
The next EU Council meeting is not scheduled to take place until 20 October, which may buy her some breathing space, as she sets up a negotiating team and establishes some "red lines" on issues such as immigration and access to the single market before official exit talks begin.
But other EU leaders are reluctant to allow Britain to dictate the terms and pace of the country's exit and Mrs May will be under pressure to trigger Article 50 as soon as possible. She will also be under pressure from some in her own party to get on with it, amid fears of backsliding on her commitment to get the UK out of the EU.
A snap general election?
Theresa May dismissed calls for a snap general election after David Cameron's exit, saying the Conservatives won a majority at the 2015 general election and her job is to continue with the work of her predecessor while pursuing fresh economic and social ideas.
As it stands, current rules mean the UK is not scheduled to go to the polls again until May 2020 but, as the last few weeks have proved, things can change rapidly in politics. Will Mrs May want to wait to go to the country in four years on the basis of the Brexit deal she has struck or will she seek a fresh mandate earlier?
Hanging over her will be the spectre of former Labour PM Gordon Brown, who also entered Number 10 without winning a general election or a vote of party members, and who torpedoed his chances of winning a mandate of his own with his will-he/won't-he wobbling over a snap election.
Visiting the rest of the UK
David Cameron used his first week in office to visit Scotland, Wales and Northern Ireland - could his successor do the same?
Scottish First Minister Nicola Sturgeon has said she wants an "early engagement" with the new UK prime minister on Europe to ensure that Scotland - which voted to remain in the European Union - is properly represented in the process of leaving and to make the case for retaining important links such as access to the single market.
While the initial discussions are likely to be cordial, there are clearly fault lines ahead given that Mrs May has said "Brexit means Brexit" and Ms Sturgeon has raised the prospect of a second independence referendum if the UK ends up outside the EU. At all levels, it will be a fascinating relationship over the coming years.
Northern Ireland has its own concerns about Brexit - focusing on security and border controls with the Republic of Ireland - which will need to be addressed while there is a separate issue in Wales about the pace of further devolution and its relationship with Westminster.
The first Commons vote - Trident?
Mrs May will face her first Prime Minister's Questions on Wednesday 20 July, always a daunting task for a new leader, but her government's first Commons test may come even earlier than that.
David Cameron, in one of his last acts as prime minister, announced there would be a vote on renewing the UK's nuclear weapons on Monday, 18 July.
Mrs May has said she will press ahead with the vote, which is likely to expose deep divisions in the already fractured Labour Party.
She has argued that replacing the UK's continuous-at-sea fleet of Vanguard submarines would send a message that the country was "committed" to working with Nato allies after voting for Brexit.
Theresa May has already indicated a break with the recent past by signalling the current government's goal of securing a budget surplus by 2020 will be abandoned.
She has also ruled out an emergency Budget in the wake of the vote to leave the EU, an idea floated by George Osborne during the referendum campaign which infuriated many Tory MPs.
Mr Osborne is unlikely to stay in the Treasury so whoever succeeds him as chancellor will set the direction of policy - but they will be under twin political and economic pressures to alleviate the impact of austerity while keeping the public finances under control.
Mrs May has said that while spending will continue to be cut, she wants to avoid tax rises, so any fiscal loosening may come through borrowing remaining higher for longer than previously planned - seen as desirable by some Tory MPs to facilitate investment in infrastructure when interest rates are so low.
The first clear signal of where the new government is heading could come in Mrs May's leader's speech to the Tory conference in October and then the chancellor's Autumn Statement, which is normally delivered at the end of November or early December.
Another decision that has been put off for years that will have to be resolved by the new prime minister.
An independent review earlier this year urged a new runway at Heathrow as the best option for the UK but the government held off, saying publicly that more environmental research was needed although critics said the further delay was sign of political indecision.
The business community is pressing for a decision which its says would be a vital signal that the UK is open for business in the wake of the uncertainty of Brexit. But there are major obstacles, not just over the issue of air quality and the cost, but political problems.
Theresa May's cabinet could be stuffed full of opponents of Heathrow expansion - such as Justine Greening, Boris Johnson and Theresa Villiers - and the new PM, whose Maidenhead constituency is under the flight path, is also thought to have had reservations in the past.
Theresa May's diary
- Wednesday 13 July: Theresa May will officially become prime minister after David Cameron tenders his resignation and Mrs May is formally asked to form a government by the Queen
- Wednesday/Friday 13-15 July: Cabinet and ministerial reshuffle expected. Who will get the big jobs?
- Tuesday 19 July: She is expected to chair her first cabinet meeting
- Wednesday 20 July: She will make her debut at Prime Minister's Questions up against Jeremy Corbyn
- 4-5 September: She will attend the G20 summit in China, expected to be her first major international conference
- 1 October: The prime minister will celebrate her 60th birthday
- 5 October: She will deliver her first leader's speech to the Conservative Party conference
- 20 October: The first EU Council meeting of May's leadership in Brussels
- November: The chancellor's Autumn Statement