The BBC Reality Check team has been checking claims made by the five remaining candidates to replace Theresa May in their live BBC debate.
Here are the verdicts on one claim from each of them in the event chaired by Emily Maitlis.
Boris Johnson on tariffs
All the candidates were asked about their plan for the Irish border after Brexit (most want to change the Irish backstop plan negotiated by Theresa May to avoid a hard border between Northern Ireland and the Irish Republic).
Boris Johnson was challenged by Rory Stewart to detail what tariffs (taxes on imports) would be charged on agricultural goods crossing the border.
He said there would be “no tariffs or quotas” because “what we want to do is get a standstill in our current arrangements under GATT 24” until a free trade deal had been negotiated.
GATT 24 is an article of the General Agreement on Tariffs and Trade. Supporters of a no-deal Brexit say it would allow the UK to continue to trade with the EU without tariffs for up to 10 years, while the two sides were negotiating a permanent future trade agreement.
But you can’t use it in this way – a trade agreement has to be agreed in principle before Article 24 can be used.
It also needs the two sides to agree – the UK can’t just impose it on the EU. You can read more about it here.
Sajid Javid on the Irish border
Home Secretary Sajid Javid outlined his plan to keep the Irish border open after Brexit – he said he would use existing technology.
Other borders between EU and non-EU countries do use technology – for example at the Sweden-Norway border cars go through unmanned border posts equipped with cameras that use an automatic number plate recognition system and goods are declared to customs before they leave warehouses.
But there is still some physical infrastructure. The EU still requires physical checks of goods at the Swedish border, so this system alone wouldn’t eliminate the need for checks at the border in Ireland – a key sticking point in negotiations.
You can read more about the question of technology on the Irish border here.
Michael Gove on schools
Michael Gove used a figure on good and outstanding schools that has been criticised by the UK Statistics Authority.
You can read the letter to the secretary of state for education here.
Jeremy Hunt on illiteracy
Foreign Secretary Jeremy Hunt was talking about schools.
“You have nearly 25% of primary school leavers unable to read – I want us to be the Conservative government that abolishes illiteracy,” he said.
While 25% of year 6 pupils in 2018 failed to meet the expected standard for reading, that does not mean they were unable to read.
Rory Stewart on emissions targets
Last week, the government said it would pass a law committing the UK to cutting net emissions of greenhouse gases to zero by 2050 (that means any remaining emissions will be offset by investing in carbon reduction projects in other countries).
Rory Stewart said that it was the most ambitious target so far set by any advanced industrial economy.
The UK would indeed be the first major industrial economy to legislate in this way, and the first G7 country to set a net-zero emissions target by 2050.
But, the Green Party has pointed out that Norway has a 2030 target for net-zero emissions, while Finland has committed to be carbon neutral by 2035.
Tory leadership race: Fact-checking the claims}