Mrs May will offer a timetable for her departure from Downing Street early next month after meeting with senior Tory MPs on Thursday 16 May. They have insisted she set a date for vacating her post.
Who is best placed to take over?
The betting favourite (odds as short as 6-4 with online gambling website Betdaq) is the MP for Uxbridge and South Ruislip.
Boris Johnson stabbed his old Etonian friend David Cameron in the back in 2016 by deciding at the last minute to back the Leave campaign ahead of the Brexit referendum.
So someone on the internet — clearly with too much time on their hands — has gender swapped Tory leadership contenders and I think more people need to see it @Conservatives #Toryleadership #Genderswap #Tory pic.twitter.com/HpUK49CRAh— Peter C. Barnes (@Barnesy19) 15 May 2019
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.
Despite all that he 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.
I've just had a horrible thought. A Tory leadership election is going to end up with the main contenders destroying each other, leaving only Chris Grayling standing because no-one thought to stab him in the back first.— Daniel Gibson (@dlsgibson) 9 April 2019
If he were to be elected as leader and become the prime minister Johnson would have to use all of his legendary charm to persuade EU leaders to change their stance on the Northern Ireland backstop at the last minute.
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.
Mrs. May hoped he would be able to drive through a compromise with the EU after Davis' prickly personality and Leave stance failed to endear him to chief Brexit negotiator Michel Barnier.
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."
"The first is that the terms being offered by the EU threaten the integrity of the United Kingdom and the second is that they would lead to an indefinite if not permanent situation where we're locked into a regime with no say over the rules being applied, with no exit mechanism," Raab said.
His position has suddenly made him a key contender for the top job, despite his lack of experience and his apparent flip-flopping.
Esther McVey, 51, was one of the first to throw her hat into the ring for the latest Tory leadership contest.
McVey, a former TV presenter from Merseyside, announced her bid on 9 May but remains an outsider with odds of between 25-1 and 66-1 being offered on her by bookmakers.
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.
Javid — who replaced Amber Rudd as Home Secretary in April 2018 in the aftermath of the Windrush scandal — has been quiet on Brexit and kept his powder dry.
He has remained a loyal acolyte of Mrs. May and he may be rewarded for that in a leadership contest as acts of betrayal rarely pay dividends in politics.
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 Javid — whose parents came to Britain from Pakistan — has to win first.
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.
Gove, 51, was born Graham, rather than Michael, to a young single mother in Edinburgh and was given up for adoption.
The Goves, who adopted him, lived in Aberdeen and his father ran a fish processing business but young Michael was sent to private school and grew up with a distinctly non-Scottish accent.
He went on to Oxford and then worked as a journalist before becoming an MP in 2005.
Hugely intelligent, Gove lacks the charisma and popular appeal of Johnson and has made plenty of enemies in the party, so he is not therefore seen as a likely winner.
In the 2016 leadership election Gove only won 48 votes and was eliminated after the second ballot.
He was widely seen to have paid the price for his betrayal of Boris Johnson.
In many ways Leadsom, 56, is a carbon copy of Theresa May and for that reason she may not be chosen as the next leader.
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 polled more votes than Gove, Liam Fox and Stephen Crabb and was all set for a head-to-head contest with Mrs. May until she made a fatal blunder.
She gave an interview to the Sunday Times in which she appeared to say that the fact she was a mother gave her the edge over Mrs. May, who is childless.
Despite claiming her remarks had been taken out of context, Mrs. Leadsom bowed out of the contest, leading to Mrs. May becoming leader and prime minister.
Mrs. Leadsom was forgiven by Mrs. May and became Leader of the Commons, where she has had the thankless task of trying to get the Brexit Bill through three times, and is set to do it all over again in the first week of June.
You don't get much more ambitious than Rory Stewart.
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 — and he has already touted himself as a contender for the leadership.
Stewart, 46, has quite 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.
Hunt was deeply unpopular with nurses and doctors — and arguably with patients — during his spell as Health Secretary.
But managing the NHS is a tough gig and Hunt was perceived within the Conservative Party as having made a good fist of it and was rewarded by Mrs. May by being promoted to Foreign Secretary when Boris Johnson resigned.
One of his first jobs was to visit China in July 2018 and while there he married a gaffe which received widespread publicity.
"My wife is Japanese — my wife is Chinese. That's a terrible mistake to make," Hunt said during an excruciating meeting in a clip that went viral.
Hunt's wife, Wang Yi, is indeed Chinese and her parents still live in the ancient city of Xian.
If Hunt were to become Prime Minister he would no doubt use all his connections to try and improve Anglo-Chinese trade.
Jacob Rees-Mogg, 49, is the chairman of the European Research Group (ERG), a wing of the Tory Party which is fiercely opposed to the EU and wants a hard Brexit.
He has been deeply critical of Mrs. May's leadership and her handling of the Brexit negotiations.
Rees-Mogg told an ERG meeting in October 2018: "The Prime Minister will lead the conservatives into the next election and if you find MPs privately who think that is a good idea in any number, I would be quite surprised."
The MP for North East Somerset, is proud of his plummy accent and his aristocratic background.
One of his children is called Anselm Charles Fitzwilliam Rees-Mogg and another is the splendidly monikered Sixtus Dominic Boniface Christopher Rees-Mogg.
He is a strong supporter of the monarchy and the Church of England and has opposed same sex marriage and abortion.
His popularity has waned considerably since 2018 and he is now a 50-1 long shot with betting websites like Bet365.
After resigning as Home Secretary in April 2018, it would be a remarkable comeback if Rudd were to become Prime Minister just over a year later.
Rudd, 55, quit after bungling the handling of the scandal of Caribbean immigrants — the so-called Windrush generation — being deported from the UK despite living and working in the country for many years.
But she showed her boomerang qualities in November 2018 when she was appointed to become Secretary of State for Work and Pensions after Esther McVey resigned over the Brexit deal.
Rudd is seen as one of Theresa May's most loyal acolytes, something which could count against her if the party's grassroots want a complete change of direction.
Another problem Rudd faces is that she represents Hastings, a marginal constituency which she only held onto last year after a recount.
If she were to become Tory leader it is more than likely she would be moved to a safer constituency, such as Mid-Sussex, rather than risk losing her seat at the next General Election.
Rudd was a Remainer before the Brexit referendum but is seen as a unifier and, despite her handling of the Windrush scandal, has many admirers within the Tory Party.
Born in the Labour heartland of Rotherham, South Yorkshire, and hailing from a family of steel workers, Greening comes from a very different background to the rest of the contenders.
She also became the first openly gay woman Cabinet minister when she came out in 2016.
Greening, 50, was promoted by Mrs. May to be Secretary of State for Education but resigned in January 2018 after reportedly refusing a sideways move to the Department of Work and Pensions.
As an MP for Putney in south-west London, she has long opposed the expansion of Heathrow Airport, a position which would be very difficult to maintain as Prime Minister.
She is a Remainer and is well-regarded for her integrity but may be unable to get enough support to beat Johnson or Raab in a leadership contest.
A complete outsider, at 66-1, the MP for Plymouth is a former British Army officer who criticised those who wear white poppies in the run up to Remembrance Sunday.
Earlier this month he banged the drum again for the armed forces when he said he would not support Mrs May's government until she offered protection for elderly veterans who faced historical prosecutions for events in Northern Ireland.
Mercer is 37 and was only elected in 2015, so perhaps his time will come further down the line.
Davidson is not even an MP — she is the leader of the Scottish Conservatives and an MSP at Holyrood.
Odds of 250-1 (down from the 500-1 being offered in November) are being touted on Farage, who quit UKIP last year and set up the Brexit Party, which is campaigning in the European elections for a no-deal Brexit.