In Washington, Barack Obama, the US president, said his administration was excited about the debate taking place in Iran as part of the elections.
"Whoever ends up winning, the fact there has been a robust debate hopefully will advance our ability to engage them in new ways," he said.
Long queues were seen at numerous voting centres and some people said they had waited more than two hours to cast their ballots.
Officials said they expected a turnout of about 70 per cent.
Observing the crowd at one polling station in Tehran, Sadegh Zibakalam, a professor of political science at Tehran University, said the official prediction looked likely to be too conservative.
"All the indications are that it will definitely be more than 70 per cent," he told Al Jazeera."
The high number of voters could approach the record of nearly 80 per cent in 1997 when Mohammad Khatami, a reformist candidate, swept the presidential election.
The four candidates for the presidency were cleared by Iran’s Guardian Council, a panel of six senior clerics and six Islamic jurists
The council, which disqualified the rest of the 475 potential candidates who registered, bars women from standing
All Iranians aged over 18 can vote, which means 46 million of Iran's more than 70 million people are eligible
If no candidate wins at least 50 per cent plus one vote of all ballots cast, including blank ones, a run-off round between the two leading candidates will be held on the first Friday after the election result is declared
A high turnout could indicate voting by many pro-reformers who stayed away from the elections four years ago, when Ahmadinejad won on a pledge to revive the values of the 1979 Islamic revolution.
"The huge turnout [this time] is an indication that maybe some Iranians regret that they didn't vote four years ago and they want to make sure that they are present at the polling station," Zibakalam said.
Crowds formed quickly at many voting sites in areas considered both strongholds for Ahmadinejad and Mousavi, who served as prime minister in the 1980s and has become the surprise hero of a powerful youth-driven movement.
Soon after voting got under way, Mousavi claimed that some of his representatives had been blocked from entering polling stations to monitor the vote.
The Reuters news agency also reported an adviser to Karroubi as saying that Ahmadinejad's opponents were concerned about a lack of ballot papers, the closure of seven reformist websites and the blocking of text messages, which they had used in their campaigns.
For his part, Ahmadinejad has accused his rivals of using "Hitler-style" smear tactics against him and said they could face jail for insulting the president.
Preliminary results are expected early on Saturday.
If none of the candidates win 50 per cent of the votes, a run-off will be held on June 19 between the two front-runners.