Theresa May and Jeremy Corbyn attempted to convince a live studio audience they should be the next prime minister in a special general election programme.
Mr Corbyn was quizzed about his foreign policy views, past campaigning in Northern Ireland and tax-raising plans.
Mrs May defended her social care reforms and was repeatedly asked if she had changed her mind on Brexit.
The leaders did not appear together, as Mrs May declined to take part in a head-to-head encounter.
The Labour leader chose to be first up in the Battle for Number 10 show, which was broadcast by Sky News and Channel 4, after winning a coin toss.
In a QA with the studio audience, a business owner attacked Mr Corbyn’s “ruthless short-sighted policies” such as increasing corporation tax and putting VAT on private school fees.
“This country is badly divided between the richest and the poorest,” the Labour leader replied, asking the man whether he was “happy” that children were going to school hungry and being taught in “supersized” classes.
Brexit and migration
Mr Corbyn was also pressed on his previous campaigning in Northern Ireland, with a man in the audience accusing him of attending a commemoration for members of the IRA, a paramilitary group.
The Labour leader said he had been seeking a “dialogue” in the 1970s and 1980s and that he had marked a minute’s silence “for everyone who died in Northern Ireland”.
He was also asked back-to-back questions on Brexit – one from a Leave voter calling for immigration controls and another from an unhappy Remain supporter.
Answering the first, he refused to set a target for migration numbers, saying Labour would act to prevent the undercutting of wages, and told the Remain supporter Labour had to “accept the reality of the referendum”.
Mr Corbyn, a committed republican, was asked by Jeremy Paxman on why abolishing the Monarchy was not in Labour’s manifesto.
“It’s not in there because we are not going to do it,” he replied, adding that he had “a very nice chat with the Queen”.
He was also challenged on his comments at the time of the 1982 Falklands War, which he called a “Tory plot”.
Mr Corbyn said he had been pressing for a “stopping” of the conflict and that the UN should have had the chance to prevent it from happening.
He said he had described the death of Osama Bin Laden as a “tragedy” because he wanted him to be arrested and put on trial.
He also defended using the term “friends” towards the militant Islamist group Hamas, calling it “inclusive language” at a meeting where he had been promoting the need for a two-state solution in te Middle East.
May on social care
Mrs May was asked about the Tories’ planned reforms to the way social care is funded, with a man in the audience describing them as a “dementia tax”.
The PM confirmed an overall cap would be put on costs, which had not been included in the Tories’ election manifesto, but did not say where this would be set.
Mrs May promised a consultation document would be published and that she would listen to charities and voters on where the cap should be, adding that the social care system would “collapse” without reform.
She would also not set a level at which the winter fuel allowance will be taken away – the Tories have said richer pensioners will no longer receive the benefit.
“I’ve met pensioners who have said they don’t think they should get that winter fuel payment,” she said.
She was repeatedly asked by Mr Paxman whether she had changed her mind on Brexit – having campaigned for a Remain vote before the referendum.
“I take the view that we can make a success of Brexit,” she said, adding that the referendum had “drawn a line” under the Remain versus Leave debate.
‘Constant work’ on migration
Having repeatedly ruled out calling a general election, she said her decision to trigger was because of rival parties “trying to frustrate the process”.
And like Mr Corbyn, she would not commit to a figure she would be prepared to sanction as the UK leaves the EU.
“It’s a question of getting the right deal for us,” she said.
On immigration, the former home secretary defended the repeated failure to hit the government’s target of reducing net migration to below 100,000.
She said there was “no single measure” to change immigration figures, describing it as a “constant work”.