It is Afghanistan’s first transfer of power via the ballot box.
There are reports of ballot paper shortages and sporadic violence from across the country.
Eight candidates are seeking to succeed President Hamid Karzai, barred by the constitution from seeking a third term.
Hamid Karzai has declared Saturday’s poll “a success”. Final results may not be declared for days.
A massive operation was launched to thwart the Taliban, who had vowed to disrupt the election, and heavy rainfall may have depressed turnout in some areas.
Independent Electoral Commission (IEC) chairman Ahmad Yousuf Nouristani said its latest estimates were that more than seven million people had voted by 17:00 local time, when the polls had officially closed and counting began.
Two-thirds of those who voted were men and one third women, the commission believes. Some polling stations stayed open until 21:00 to allow everyone queuing to vote.
“This election was a message to the enemies of Afghanistan,” Ahmad Yousuf Nouristani said.
“With this determination of the honorable people of Afghanistan, the enemies were defeated.”
IEC secretary Ziaul Haq Amarkhel, asked to comment on widespread reports of polling stations running out of ballot papers, said this information was “false”.
Earlier there were reports of polling centers running out of ballots hours before the polls closed in many areas, including Kabul, northern Takhar province, north-eastern Badakhshan province, eastern Paktia province, and Nimroz province in the south-west – where one man, Abdul Ahad, said he and 15 family members had been to every polling centre in their district in an attempt to vote, but all of them had run out of ballot papers.
The biggest military operation since the fall of the Taliban in 2001 was rolled out for the vote. All 400,000 of Afghanistan’s police and soldiers were said to be on duty for the election.
Traffic was prevented from entering the Afghan capital from midday on Friday, with police checkpoints erected at every junction.
However, in parts of the capital voters could be seen queuing an hour before polls opened and there was a good-natured, almost carnival atmosphere, with many people on the streets.
Across the country, 10% of stations were declared unsafe to open by the election commission.
The Afghan ministry of defense said three major incidents had taken place on polling day.
Fears of fraud, which have marred previous polls in Afghanistan, resurfaced with reports from the southern province of Kandahar that police were preventing voters and observers from reaching polling stations.
The interior ministry said two police officers were arrested in Wardak province for stuffing ballot boxes.
Concerns were also raised before the poll about the possible presence of “ghost” polling stations as well as the fact that the number of election cards in circulation appeared to be vastly more than the number of registered voters.
Speaking after the polls closed, Hamid Karzai said: “Despite the cold and rainy weather and possible terrorist attack, our sisters and brothers nationwide took in this election and their participation is a step forward and it is a success for Afghanistan.”
President Barack Obama, in a statement issued by the White House, said: “We commend the Afghan people, security forces, and elections officials on the turnout for today’s vote – which is in keeping with the spirited and positive debate among candidates and their supporters in the run-up to the election.
“These elections are critical to securing Afghanistan’s democratic future, as well as continued international support.”
There are eight candidates for president, but three are considered frontrunners – former foreign ministers Abdullah Abdullah and Zalmai Rassoul, and former Finance Minister Ashraf Ghani Ahmadzai.
Dr. Abdullah Abdullah has fought a polished campaign, Ashraf Ghani has strong support among the new urban youth vote and Dr. Zalmai Rassoul is believed to be favored by Hamid Karzai.
However, no candidate is expected to secure more than the 50% of the vote needed to be the outright winner, which means there is likely to be a second round run-off on May 28.
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