The BBC will hold two televised debates with candidates vying to become the next prime minister.
Ten Conservative MPs have entered the race to become both party leader and PM so far, after Theresa May announced she would stand down on 7 June.
All candidates in the race by mid-June will be invited to a hustings event on BBC One and the final two will go head-to-head in a Question Time Special.
Mrs May urged her successor to seek a consensus on Brexit in Parliament.
The prime minister spoke at a summit of EU leaders in Brussels, where senior figures reiterated that the agreement they secured with the UK could not be reopened.
Some of the leadership contenders have indicated they are willing to countenance leaving the EU on 31 October on a no-deal basis - but others say they feel that would be unacceptable.
Mrs May said it was important to deliver on Brexit - and "I think the best way to do that is with a deal."
Who is standing?
The declared candidates to replace Mrs May are:
- Environment Secretary Michael Gove
- Health Secretary Matt Hancock
- Foreign Secretary Jeremy Hunt
- Home Secretary Sajid Javid
- Former Foreign Secretary Boris Johnson
- Former Leader of the House Andrea Leadsom
- Housing minister Kit Malthouse
- Former Work and Pensions Secretary Esther McVey
- Former Brexit Secretary Dominic Raab
- International Development Secretary Rory Stewart
How does the contest work?
The deadline for Tory MPs to put their names forward for the contest is the week commencing 10 June, and they must have at least two of their colleagues supporting them.
Once the deadline passes, there will be a series of votes by MPs on which candidate they prefer.
Each round will see the MP with the least number of votes struck off the list, until there are only two candidates left.
These two MPs will then be voted on by members of the Conservative Party to choose the next leader - and next PM.
What have the candidates said so far?
Despite the deadline still being a week away, the runners and riders have already begun making their pitches to the public.
- Foreign Secretary Jeremy Hunt warned his party will be committing "political suicide" if it tries to push through a no-deal Brexit.
- But Work and Pensions Secretary Esther McVey, disagreed, arguing the real "political suicide" would be if the UK did not leave the EU by Halloween.
- Housing Minister Kit Malthouse - the latest to join the race - said: "Those people who say no deal would be a catastrophe and those people who say it would be a walk in the park are both wrong - it is somewhere in the middle."
- Elsewhere, Health Secretary Matt Hancock welcomed the decision to hold TV debates, arguing the "future of the country" was at stake.
- Former Brexit Secretary Dominic Raab released a video saying he would focus on "fairness" - from cutting taxes for the lowest paid and lowering the cost of living, to increasing opportunities for young people.
- International Development Secretary Rory Stewart is carrying out a "listening exercise" on Brexit by inviting people over Twitter to come and meet him in a number of locations.
What about the EU? And Theresa May?
Mrs May said she would not comment on the views of those trying to succeed her as prime minister, but said the Tories' dismal showing in the European elections should send a clear message.
"I think what it shows is the importance of actually delivering on Brexit. I think the best way to do that is with a deal," she said.
"But it will be for my successor and for Parliament to find a way forward and get a consensus."
A succession of leadership candidates have said they will go back to Brussels and attempt to renegotiate a better withdrawal deal, but on Tuesday, EU leaders once again insisted that would not happen.
"There will be no renegotiation," said European Commission President Jean-Claude Juncker.
Elsewhere on Tuesday, Commons Speaker John Bercow insisted MPs would get a say on whether the UK leaves without a deal.
He dismissed claims from some Brexiteers that because a no-deal exit is the default position in law, it will "inevitably" happen if no agreement is reached by then between the next PM and Brussels.
Why have the televised debates?
Fran Unsworth, director of BBC News and Current Affairs, said: "The decision being made by Conservative Party members will profoundly affect us all, so it feels right that BBC audiences get a chance to see the candidates debate with each other, and that we scrutinise the various policy proposals they will be standing on.
"Although the final say will fall to Conservative Party members, it's firmly in the public interest for audiences to question and hear from the next potential prime minister."
The BBC programmes will be shown in June, with dates and times to be confirmed.
The last two runners in the Tory leadership contest will also be invited to take part in one-on-one interviews with the BBC's Andrew Neil.
And Sky News has announced it is organising a live head-to-head debate between the final two candidates for the Tory leadership with a studio audience made up of Conservative voters.