Jeremy Corbyn has said that he won’t lead Labour into the next election, after the party suffered its worst defeat since 1935.
Current shadow chancellor John McDonnell has ruled himself out of the contest, and says he wants the party to pick a female leader.
So far there are more women than men included in the possible runners and riders in the race.
Here are some of the potential hopefuls, with analysis from BBC Reality Check.
Who will run for Labour leadership?
The 40-year-old MP for Wigan has said she is “seriously considering” standing for the leadership. One of a clutch of shadow ministers who resigned from Jeremy Corbyn’s frontbench after the Brexit referendum, she has been urging her party to concentrate on winning support in smaller towns.
The 57-year old shadow Brexit secretary has also said he is “seriously considering” running. He is likely to be seen as the centrist candidate in the race. A passionate Remainer, he was director of public prosecutions before entering Parliament.
The 40-year old shadow business secretary is another MP to announce she is considering going for the top job. One of a new generation of MPs on the left of the party who is close to Mr Corbyn’s inner circle, she represented Labour in a TV debate during the election.
The 38-year old Birmingham Yardley MP became the third Labour figure to declare she is standing for the leadership. She has been one of the most outspoken critics of Jeremy Corbyn’s leadership and the party’s record on tackling anti-Semitism, bullying and harassment.
The 59-year old shadow foreign secretary was the first to declare she is running to succeed Mr Corbyn. She deputised for him at Prime Minister’s Questions, but was replaced after publicly calling for Labour to back another EU referendum.
The 50-year old, a former cabinet minister under Gordon Brown’s premiership, said last month that she would “decide over Christmas” whether to stand, and has made no statement since then. She was an unsuccessful challenger to Jeremy Corbyn during the 2015 leadership contest.
The 48-year-old shadow Treasury minister resigned from the party’s frontbench last year in order to oppose the bill triggering the Brexit process. An early supporter of Jeremy Corbyn, he rejoined in January last year.
What about the deputy leader?
The post was vacated at the election when former Labour MP Tom Watson said he was stepping down, both from the role and as a member of Parliament.
This means it will also be up for grabs in the upcoming contest.
Shadow justice secretary Richard Burgon announced on Twitter he would be running for the deputy leader post “after a break and discussions with MPs and party members”.
He previously said he had “made no secret” of the fact he would like Ms Long-Bailey to be the next head of the party.
But the shadow business secretary confirmed she would back Angela Rayner to be second in command.
The 39-year-old shadow education secretary was a care worker and Unison official before becoming an MP, and has described herself as being on the “soft left” of the party.
When could it happen?
Jeremy Corbyn has said that it’s up to the National Executive Committee (NEC) – Labour’s governing body – to decide when he goes as leader.
But he has said that he expects a new leader to be selected early in the new year and the NEC are expected to meet next week.
The party’s rulebook says that when both the leader and deputy leader are “permanently unavailable”, the NEC calls a postal ballot.
The NEC may want to have a new leader in place before local elections in England, scheduled for 7 May.
In 2015, the process took more than four months. Ed Miliband resigned on 8 May and Jeremy Corbyn was announced as winner on 12 September.
Who can run?
Candidates for leader and deputy leader have to be MPs, and they need nominations from 10% of Labour MPs and MEPs.
And in a new rule, candidates also need nominations from 5% of Labour’s constituency parties.
Alternatively, they need nominations from three affiliated bodies, two of which must be trades unions, adding up to 5% of affiliated members.
Who can vote?
Members of the Labour Party, affiliated trades unions (if they opt in), and socialist societies such as the Fabians, all get one vote each.
In 2015, non-members were allowed to register as supporters and vote in the contest for a £3 fee.
Those new registered supporters voted overwhelmingly for Jeremy Corbyn, though he gained enough support from members and affiliates to win anyway.
In 2016, when Owen Smith challenged Jeremy Corbyn, the cost of registering was raised to £25 and people were given only two days to sign up.
The cost and the time period for registering this time will be in the hands of the NEC.
How does the vote work?
The votes are cast on a one-member, one-vote basis, by preferential ballot.
That means that voters rank the candidates in order of preference. If any candidate gets more than half the votes, they win.
If not, the candidate with the fewest votes is eliminated, and their second preference votes are redistributed.
If that results in any candidate with more than half the votes, they win. If not, the remaining candidate with the fewest votes is eliminated and their votes redistributed, until the contest produces a winner.
Who controls the process?
Labour’s National Executive Committee has 39 members, representing the trades unions, the shadow cabinet, Labour’s elected representatives at local, national and European level, and constituency parties.
Supporters of Jeremy Corbyn and the grassroots campaign group Momentum are strongly represented on the NEC, and they are likely to use their influence to promote a left-wing candidate in the coming election.
What about the deputy leader?
There is also a vacancy for deputy leader, as the incumbent Tom Watson stood down. It would be up to the NEC to decide when to have an election for a new deputy, although it would be cheaper and simpler to hold the votes at the same time.