KABUL, Afghanistan — Afghans flocked to polling stations nationwide on Saturday, defying a threat of violence by the Taliban to cast ballots in what promises to be the nation’s first democratic transfer of power. The turnout was so high that some polling centers ran out of ballots.

The excitement over choosing a new leader for the first time appeared to overwhelm the fear of bloodshed in many areas, as Afghans embarked on a major transition nearly 13 years after the U.S.-led invasion toppled the rule of the Taliban.

President Hamid Karzai, the only leader the country has known since the Islamic movement was ousted, is on his way out, constitutionally barred from a third term. International combat troops are leaving by the end of the year.

And Afghans are left largely on their own to face what is likely to be an intensified campaign by the Taliban to regain power.

The vote is the first for Afghans in which the outcome is uncertain. Voters are choosing from a field of eight presidential candidates, as well as selecting provincial council members. With three front-runners in the presidential race, a runoff was widely expected since none is likely to get the majority needed for an outright victory.