As a minority government, the Conservatives cannot push through the Brexit they want.
The EU has said it will not renegotiate a new deal, but trying to force a no-deal Brexit through parliament could result in a constitutional crisis, with MPs repeatedly voting to block such a move.
The next prime minister will be reluctant to call a general election because the Tory party might well lose, after coming a poor fifth in the European elections. Frontrunner Boris Johnson has already ruled out the option.
One other possibility is a second referendum on all options including remaining in the EU, but the Tories are against the move, which would spark fury from many Brits after a majority voted to leave the EU three long years ago.
The new PM is set to face an immediate confidence vote from Labour to establish whether they have the support of the House of Commons. Mrs May has a working majority of just five after a deal with the Democratic Unionist party, so the new leader could easily fail to form a government.
It is known that many MPs have reservations about Mr Johnson.
Leadership contestant and former Brexit secretary Dominic Raab refused to rule out asking the monarch to suspend Parliament if he wins the race, which formally begins today.
But other candidates tore into the idea, with international development secretary Rory Stewart calling it "as disturbing as it is ludicrous" and health secretary Matt Hancock writing to all contestants urging them to avoid the extreme step.
"To suspend parliament explicitly to pursue a course of action against its wishes is not a serious policy of a prime minister in the 21st century," he wrote. "What kind of message would this send around the world about our values when so many have given so much for the rights of democratic freedom?"
Mel Stride, a supporter of environment secretary and leadership candidate Michael Gove, echoed the feelings of many MPs when he said: "Her majesty should be kept out of the politics of our Parliament."
John Bercow, the Speaker of the House of Commons, has delayed his exit until Brexit is resolved, irritating Tory MPs who believe he is biased.
He blasted the suggestion parliament could be suspended, saying: "That is simply not going to happen. It is just so blindingly obvious that it almost doesn't need to be stated, but apparently, it does."
Mr Johnson, Mr Gove, Mr Hancock, Sajid Javid, Jeremy Hunt, Andrea Leadsom and Mr Stewart have all ruled out the idea of "proroguing" parliament during early hustings. They have all committed to making Brexit by October, rather than further drawing out the painful process, as many European MEPs fear will happen. How they will achieve this is far from clear.
Mrs May will today privately exchange letters with Conservative Party chairman Brandon Lewis and the leadership battle will begin. It is sure to be a bitter one.
