The Conservatives suffered heavily in last week's European election, with huge swathes of Tory voters defecting to Nigel Farage's Brexit Party, which campaigned for a no deal Brexit.
The Tories will hold a leadership contest next month and there are now ten MPs in the running for the position.
Who are they and what do they stand for?
Boris Johnson
Boris Johnson, the Conservative MP for Uxbridge and South Ruislip, has managed to rebuild his reputation after his disastrous leadership bid in 2016.
He stabbed his old Etonian friend David Cameron in the back by deciding at the last minute to back the Leave campaign ahead of the Brexit referendum.
He then stood for the leadership after Cameron resigned in the wake of the referendum vote but dramatically quit the race after his ally Michael Gove suddenly turned on him and publicly doubted his credentials.
Johnson remains popular with grassroots Conservative Party members and his reputation with them was not necessarily tarnished by his spell as Foreign Secretary despite a string of gaffes.
Johnson, who will be 55 next month, eventually resigned from the Cabinet in September 2018.
"We are truly headed for the status of a colony — and many will struggle to see the economic or political advantage of that particular arrangement," he said in a resignation letter calculated to damage May and her Brexit negotiations.
Johnson's pro-Brexit credentials are expected to endear him with grassroots Tory members in the shires.
He has not given any broadcast interviews since May announced she would be stepping down.
Commenting in the Daily Telegraph on the Tories' disastrous European election results, where they were wiped out by the Brexit Party, Johnson wrote: "After these dismal elections, we need a proper Brexit and a fresh agenda to reunite the country."
It would not be a great leap of imagination to see Johnson's "fresh agenda" including support for a no deal Brexit.
But Parliament has already ruled out a no deal Brexit so Johnson, as Prime Minister, would have to either call a general election in the hope of getting enough MPs elected who would support it, or call a second referendum with a straight choice — no deal Brexit or remaining in the EU.
Bookmakers' odds — 6/4 (Ladbrokes)
Michael Gove
Michael Gove, the Environment Secretary, has pledged free British citizenship for three million EU nationals after Brexit if he becomes prime minister, according to The Guardian.
Gove is also reportedly willing to abolish the burden of providing proof of settled status.
Conservative MP Alberto Costa, who is of Italian origin, told the Guardian: "This is Michael's way of saying to EU nationals: I'm sorry, the Vote Leave campaign was never about EU citizens' rights. He is unambiguously demonstrating that leaving the EU is not about moving the goalposts of citizens' rights and accordingly, he has my full support as a candidate for the premiership."
Gove was a passionate supporter of the Leave campaign in the run up to the 2016 referendum but the move, which he is expected to announce later on Tuesday 28 May, is thought to be an acknowledgement that he is seen by many as dogmatic and uncaring.
Gove has considerable intellectual clout but may not have the popularity among ordinary members to be elected leader, but he can expect to be given a big job — possibly Chancellor or Home Secretary — in Cabinet under the new leader.
Bookmakers' odds — 4/1
Dominic Raab
Handsome and softly spoken, Raab is a genuine contender to replace May and could also be the candidate for those who believe the next Prime Minister should be"anyone but Boris".
In 1990 Michael Heseltine was the main challenger to Margaret Thatcher but he had so many political and personal enemies in the party that he eventually lost out to John Major, who was seen as bland and harmless but basically Thatcherite.
It is quite possible Johnson will lose out to Raab in a similar scenario.
Raab, 45, was a relatively unknown backbencher until he was appointed Secretary for Exiting the European Union in July 2018 following the resignation of David Davis.
Raab's constituency, Esher and Walton in Surrey voted 58.4 percent to 41.6 percent to remain in the European Union but he has not let that stop him speak out in favour of Brexit.
He was heavily involved in drafting the 585-page document which was finally agreed by May's Cabinet in November 2018.
But the following day he quit, saying the deal had "two major and fatal flaws."
He has remained opposed to May's deal ever since and, despite his lack of experience and his apparent flip-flopping, Raab may be in a good position to get the top job if he goes one step further and supports a no deal Brexit.
Bookmakers' odds — 4/1
Jeremy Hunt
Jeremy Hunt has warned his party will be committing "political suicide" if it tries to push through a no deal Brexit.
The foreign secretary knows his chances of getting May's job are slim, so his comments in the Daily Telegraph and on the BBC may have been made on the basis that he had nothing to lose.
Hunt said if the Tories supported a no deal Brexit it would lead to a general election, which could see Labour take power.
Hunt said: "If you go to (Brussels) with the right tone, with a deal that recognises their legitimate concerns over the border of Ireland, if you go with a negotiating team that gives them the confidence you can deliver with a majority in the House of Commons, you have the prospect — I don't say the guarantee — but you have the prospect of getting a deal."
The former Health Secretary's conciliatory tone is not expected to go down well with Conservative Party members, who are generally old, fed up of the EU negotiations and in favour of a no deal Brexit.
Bookmakers' odds — 12/1
Andrea Leadsom
When Andrea Leadsom resigned as Leader of the Commons on Wednesday, 22 May, it was the beginning of the end for May, who resigned 36 hours later.
Born in affluent Buckinghamshire and educated at a grammar school, Leadsom attended Warwick University before entering the world of banking and was a fund manager at Invesco Perpetual before becoming an MP.
She backed the Leave campaign in the 2016 referendum and when Cameron fell on his sword she emerged as a dark horse contender for the leadership.
Leadsom was on the two-women shortlist for the leadership back in 2016 but she put her foot in it when she appeared to say she was more suited to be Prime Minister because, unlike May, she had children.
She promptly stepped down, allowing May a walkover, and has been a loyal supporter of the Prime Minister and her deal right up until last week.
Leadsom tweeted on Monday, 27 May, about having a "three step plan" for how to deliver Brexit which she said she would be discussing during the leadership campaign.
She said once Brexit was sorted it would allow the UK government to concentrate on other important issues like "climate crisis and social care".
That message might play well in a general election but the priority is winning the votes of the 100,000 Conservative Party members and Leadsom's success in doing so depends on the detail of her "three step plan" on Brexit.
Bookmakers' odds — 12/1
Sajid Javid
Javid — who replaced Amber Rudd as Home Secretary in April 2018 in the aftermath of the Windrush scandal — has kept his powder very dry on Brexit.
He has remained a loyal acolyte of May, but when it comes to the leadership ballot he may be seen as too similar too her and unlikely to break the Brexit stalemate.
If he were to be elected as leader the Conservative Party would claim a major first — he would be the first British prime minister from an ethnic minority.
Considering ethnic minorities traditionally vote for Labour, that would send a big signal to black and Asian voters.
But the trouble is the vast majority of Tory members are white and elderly and Javid — whose parents came to Britain from Pakistan — may find it a struggle overcoming some of their prejudices.
He studied politics and economics at Exeter University, where he was known as an admirer of Margaret Thatcher.
He became a Vice President at Chase Manhattan Bank at the tender age of 25 and later moved to Deutsche Bank in London and worked on emerging markets.
Javid, 49, is a rich man. He reportedly earned up to £3 million (US$4.1 million) a year during his 20-year banking career and now owns a £4 million (US$5.5 million) home in Fulham, a £3 million (US4.1 million) home in nearby Chelsea and sends his children to private schools.
Bookmakers' odds — 20/1
Rory Stewart
The International Development Secretary, Rory Stewart, has promised a radical "listening exercise" on Brexit.
He appeared on ITV's Good Morning Britain on Tuesday, May 28, and suggested forming a "citizens' assembly" to hammer out a Brexit deal.
Stewart explained: "It is basically a jury of 500 ordinary people, they would sit seven days a week, hear from the experts and be given evidence, and would discuss the details of Brexit. This would be people from all parts of the country, just like a jury, in order to represent the country. They would make recommendations back to Parliament."
It might be a radical idea and might have worked if suggested in 2016, but in 2019 time is of the essence and a 500-person "citizens' assembly" sounds an awful lot like a talking shop, when the UK really needs decisive leadership.
Stewart, 46, can be forgiven for being naive. He has only been in the Cabinet for a few weeks — promoted to International Development Secretary to replace Penny Mordaunt, who was bound for the Ministry of Defence.
He has an interesting biography — born in Hong Kong, and brought up in Malaysia, he spent five months in the Black Watch regiment as a second lieutenant during his gap year before reading Philosophy, Politics and Economics at Oxford University.
While at Oxford he was a private tutor to Prince William and his brother Harry during the summer.
He joined the Foreign Office and spent time as a diplomat in East Timor, Kosovo, Montenegro, Iraq and Afghanistan.
Stewart was elected as the MP for Penrith, in Cumbria, in 2010 and has risen through the ranks under Cameron and May.
But it would be a huge shock if he were to avoid being knocked out in the early rounds of the leadership contest.
Bookmakers' odds — 20/1
Matt Hancock
Matt Hancock, who has been Health Secretary since July 2018, has not made any pledges about Brexit and seems to be trying to steer a cautious course, similar to May.
Hancock said: "We've lost many long-standing voters to the Brexit Party — not because of any details of the deal we proposed — but simply because we haven't delivered Brexit yet. It's not that most people want a No Deal Brexit, but because so many rightly think we should have left by now and want us to get on with it. So it is mission critical we deliver Brexit before any general election."
Before entering politics, he was an economist at the Bank of England, and was Chief of Staff to the Shadow Chancellor of the Exchequer.
An avid sportsman, Hancock won a horse race at Newmarket races in 2012 and once played the most northerly game of cricket on record, deep in the Arctic Circle.
Bookmakers' odds — 25/1
Esther McVey
In response to Jeremy Hunt's comments, a rival contender Esther McVey said the only "political suicide" would be not leaving the EU on 31 October.
McVey, 51, was one of the first to throw her hat into the ring for the latest Tory leadership contest.
A former TV presenter from Merseyside, she announced her bid on 9 May but remains an outsider.
McVey lost her seat at the 2015 general election but returned two years later when she was parachuted into the safe seat of Tatton in Cheshire, formerly held by the Chancellor George Osborne.
In January Ms. McVey described herself as a "compassionate Conservative", to widespread derision. Her stewardship of the Universal Credit programme has been lambasted by MPs and benefits claimants, and she was forced to admit some UK citizens would be worse off under the new system.
Bookmakers' odds — 50/1
Kit Malthouse
The only one of the candidates who has never been in the Cabinet is Kit Malthouse, whose loftiest position was Deputy Mayor of London during Boris Johnson's reign.
Malthouse, a chartered accountant, served as Deputy Mayor for Policing 2008 to 2012 and Deputy Mayor for Business and Enterprise from 2012 to 2015.
Malthouse, quoted in The Sun, said: "This leadership campaign cannot be about the same old faces, scarred by the wars that have split the Tory Party over the last 3 years. I believe I'm the new face, with fresh new ideas, from a new and talented generation."
Malthouse is fairly moderate on Brexit.
He gave his name to the Malthouse Compromise — a proposal drawn up by backbenchers from Leave and Remain wings of the Tory Party, which would have implemented May's Brexit deal with the backstop replaced by alternative arrangements.
But it failed to garner enough votes and does not augur well for his chances of winning the leadership.
Bookmakers' odds — 100/1