Millions of Afghans have defied the Taliban and turned out to vote in the weekend's presidential election.

World leaders have praised the courage of Afghan voters, who cast their ballots in unexpectedly large numbers.

There is a sense of relief across Afghanistan as election day passed without any major attack, despite the Taliban's threat to do whatever it can to disrupt the poll.

"We are so happy that the election took place in a peaceful environment," says one local.

"There was good security around the country and everywhere was peaceful."

An estimated seven million Afghans voted; more than 30 per cent were women - a much better-than-anticipated turnout.

Afghan MP Shukria Barakzai is among leaders praising their people for defying the extremists.

"The way the Afghans were participating in the election - the turnout of the people in the polling stations, women and men in bad weather conditions, in an extreme high security threat - that was a fantastic slap in the face of enemy of Afghanistan," she said.

Counting the votes is expected to take six weeks.

It could be the start of a potentially dangerous period for Afghanistan at a time when the war-ravaged country desperately needs a leader to stem rising violence as foreign troops prepare to leave.

Voters are hopeful that results will be established quickly.

"We want the election result to be finalised in the first round," says one.

"Our people, government, and economy are very weak.

"If it goes to the second round, it will be a challenge for our security forces."

That challenge was underscored on Sunday in the northern province of Kunduz, when a roadside bomb hit a truck carrying ballot boxes as they were being taken to a counting centre.

Three people onboard were killed.

Thomas Ruttig, a veteran Afghan watcher with the Afghanistan Analysts Network, says one aspect of election day that is troubling are reports that many voting booths ran out of ballot papers.

"In recent elections have sometimes been the sign that there has been some tampering, which we cannot exclude," he said.

"Maybe also, if there's an unexpected turnout, the problem is that you don't have a voter registry in Afghanistan, where the people are linked to certain polling stations.

"So then people start driving around and looking for a polling station which still has ballot papers and so on."

The three main frontrunners to succeed Hamid Karzai as President are Abdullah Abdullah, Ashraf Ghani and Zalmai Rassoul.

If none of these candidates get more than 50 per cent of the vote, a run-off election between the two top vote-getters will be held later this year.