Here John Rentoul looks at the factions within the government, the bookmakers on ministers taking the top job and the percentage of support they have among Conservative Party members.
The Brexity bunch
Michael Gove, environment secretary
Odds on becoming PM: 12/1 Party membership support: 4%
Did a better job than the PM of selling her deal in the Commons this week. Can’t risk plotting after doing in his friend David Cameron and his ally Boris Johnson in quick succession. But reported to be trying to persuade Penny Mordaunt to run on a joint ticket – even though she still has her eye on the top job for herself.
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Penny Mordaunt, international development secretary
Odds: 25/1 Support: 1%
Seen as the hardest-line Brexiteer in the cabinet, more so than Gove, but a weak performer on TV.
Liam Fox, Chris Grayling, Stephen Barclay, Natalie Evans (Baroness Evans, leader of the Lords)
The cabinet ministers who voted Leave in 2016; probably happiest in theory with Gove as leader, but could live with Javid or Hunt.
The pragmatic mainstreamers
Sajid Javid, home secretary
Odds: 13/2 Support: 8%
Attracting hostility as the cabinet frontrunner. An enemy told journalists he referred to himself in the third person as The Saj; and Philip Hammond, the chancellor, caught him out in cabinet by pointing out a no-deal Brexit, which Javid had hinted at supporting, would stop him sending migrants back across the Channel.
Jeremy Hunt, foreign secretary
Odds: 9/1 Support: 6%
The other candidate often named by mainstream Tory MPs and ministers, he too has been mocked for his obvious ambition, comparing the European Union to the Soviet Union at party conference.
Gavin Williamson, defence secretary
Odds: 50/1 Support: less than 1%
Another minister whose ambition has been embarrassingly obvious, but in his case seems to have outrun his prospects.
David Lidington, Matt Hancock, Damian Hinds, David Gauke, James Brokenshire, David Mundell, Alun Cairns, Karen Bradley, Jeremy Wright, Brandon Lewis
The largest group, 10 out of the 23 cabinet members. Mostly realistic about their own prospects (although Hancock may harbour long-term ambitions). Keen to avoid the leadership falling into the hands of Johnson or Rees-Mogg.
The soft Brexit brigade
Amber Rudd, work and pensions secretary
Odds: 18/1 Support: 5%
Vigorous and articulate May loyalist but also an unapologetic Remainer. May be waiting to throw her support behind another candidate (even Johnson?) in return for becoming first female chancellor.
Philip Hammond, Greg Clark
Now pushing hard to persuade the prime minister to extend Article 50 rather than to allow a no-deal Brexit: would prefer Rudd as leader but could settle for Javid or Hunt.
And they are all under pressure from outside the cabinet…
The hard-Brexit outsiders
Odds: 13/2 Support: 20%
Most popular with party members, but unpopular with fellow MPs, having failed to use the freedom of the back benches to set out a credible alternative Brexit policy. That means he may not make the shortlist of two, chosen by MPs.
Odds: 9/1 Support: 7%
Short-lived Brexit secretary and therefore a fairly unmarked candidate with few enemies.
Odds: 25/1 Support: 15%
Another darling of the grassroots with even less chance of being chosen by fellow MPs for the final run-off.
Odds: 25/1 Support: 8%
Long-serving Brexit secretary seen as a hard Brexiteer but also one who achieved little in his two years at the new department.
The Independent has launched its #FinalSay campaign to demand that voters are given a voice on the final Brexit deal.