The result of the election so far is surprising. Reformist candidate Massoud Pezeshkian, who had been ahead in earlier results, is now trailing at just over 40%.
Mr Pezeshkian, a former heart surgeon and health minister, has promised a different approach, saying the actions of the morality police, who enforce strict dress codes on women, are “immoral”.
The run-off vote will happen if no candidate wins more than 50% of all ballots cast in this first round.
The vote is to replace former president Ebrahim Raisi, who died on 19 May when the helicopter he was in crashed into a mountain, with seven other people also killed.
Though there are 61.5 million eligible voters in Iran, turnout is expected to be low for this election. It hit record lows in parliamentary elections in March and the last presidential election in 2021.
Supreme leader Ayatollah Ali Khamenei, who is the ultimate authority in the country, has called for “maximum” turnout.
Iran was shaken by a huge wave of protests in 2022 following the death of 22-year-old Mahsa Amini, who was detained by the morality police for allegedly violating Iran’s strict dress code.
Human rights groups say hundreds were killed in the crackdown and thousands detained.
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