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Across Bosnia’s battle-scarred landscape, voters accustomed to firing bullets at one another during their civil war instead cast ballots Saturday in an election that was remarkably free of serious acts of violence.

There were 53,000 NATO troops and thousands more police officers fanned out to protect the 4,000 polling places, from Sarajevo to remote mountain villages. The monitors reported only relatively minor infractions and said voters generally conducted themselves in an orderly fashion.

“The situation has been surprisingly calm,” said Maj. Brett Boudreau, spokesman for NATO’s Peace Implementation Force.

“We are holding our collective breath but are beginning to exhale a bit with no serious incident to report.”

First election results are expected late Sunday or early Monday, but the outcome will not be entirely clear before Tuesday or Wednesday because of logistical difficulties in counting votes.

Many polling stations may experience problems in delivering ballot boxes to counting points because recent heavy rains have made it difficult to cross swollen rivers in places where bridges were blown up during the war, officials said.

Many voters were refugees who returned to their home towns for the first time in years. There were fears expressed that the refugees, bearing bitter memories of ethnic cleansing and other atrocities, could become involved in violent clashes with those who have taken over their homes.

That apparently did not happen. The refugees were bused to polling stations and then quickly bused out again after voting. They had no opportunity to see the houses where they used to live.

Election officials used 730 buses to transport refugees from the Muslim-Croat federation into Serb areas, and 55 buses to bring refugees from the Bosnian Serb republic into the federation.

Officials estimated 60 percent to 70 percent of registered voters turned out. They said some refugees were afraid to cross ethnic boundary lines to vote.

Former Bosnian Serb President Radovan Karadzic, indicted for war crimes, was among those voting in the Bosnian Serb capital of Pale. NATO troops are committed to arresting him on sight, but a spokesman said he was unaware that any NATO troops saw Karadzic vote.

In Sarajevo, the governing Muslim Party of Democratic Action announced it was asking that the election results in the Bosnian Serb republic be nullified on grounds of unspecified irregularities.

The election came nine months after the signing of the Dayton peace agreement. It was designed to implement new political structures and set the stage for a pullout by NATO troops in December. Continuing conflicts have made it all but certain that a reduced NATO force including U.S. troops will have to remain for another year or more.

Former U.S. diplomat Richard Holbrook, one of the architects of the Dayton agreement, headed a U.S. observer delegation that toured voting places on Saturday.

Holbrook said it was evident that if the entire NATO force were to pull out in December, “the place will implode.” He declined to say if U.S. troops should be part of the contingent that stays.

The American delegation was among observers from around the world sent to determine if voting was free and fair. Some organizations, such as the International Crisis Group, said weeks ago that free and fair elections were impossible because of widespread abuses on all sides, the absence of a free press, and curbs on freedom of movement that have prevented refugees from returning to their former homes.

The Organization for Security and Cooperation in Europe (OSCE), which supervised the elections, is expected to certify on Sept. 25 that they were held in free and fair conditions.

Under the Dayton terms, that will be followed by formal end to the United Nations sanctions against Yugoslavia and the Bosnian Serb republic. The sanctions already have been suspended.

Some critics of the election say they fear the Bosnian Serb republic will illegally declare its independence from Bosnia once sanctions are gone.

Holbrook said if that happened, the Bosnian Serbs would pay a price in terms of loss of economic assistance, international isolation and other punitive measures he declined to specify.

Biljana Plavsic, the acting president of the republic, came close to declaring an intention to become independent during the election campaign and was forced by OSCE officials to go on Bosnian Serb television Friday night and read a statement of apology they drafted for her.

Had she refused to read the statement, officials would have removed some candidates of her Serb Democratic Party from the ballot.

The huge NATO and police deployment on Saturday may have helped prevent serious violence, but the legacy of war was apparent in small incidents.

The most serious, reported by American observers, occurred in the Bosnian Serb town of Kupace, where Bosnian Serb officials kept one polling station designated for Muslim refugees closed. Hundreds of refugees finally gave up the attempt and went home.

Bosnian Serb authorities at one point closed the Brcko bridge in eastern Bosnia to all but pedestrian traffic, impeding the return of refugee voters from Croatia, but they reopened the bridge to bus traffic at the OSCE’s insistence.

South of Mostar, Croats stoned buses and shouted threats at Muslim voters. The Muslim police chief in the Sarajevo suburb of Ilidza ordered his men to set up an illegal checkpoint to hinder returning Serb refugees. NATO troops removed the checkpoint.

In Muslim-controlled Bugojno, Croatian poll watchers withdrew after someone threw a grenade at the home of the Croat local election commission chairman. But they later returned to their duties after being assured of NATO protection.

In the Sarajevo district of Novi Grad, voters angry at delays in the balloting procedure threw rocks and broke a plate-glass window at a polling station.

There were 24 parties contesting the election but just three that are likely to predominate: the Party of Democratic Action of Bosnian President Alija Izetbegovic in Muslim areas, the Serb Democratic Party formerly headed by Karadzic, and the Croat Democratic Party of Bosnia-Herzegovina allied to Croatian President Franjo Tudjman.

An estimated 2.9 million people were eligible to vote in the election. Almost half were refugees, and those living outside Bosnia cast absentee ballots earlier in the week. Officials said 80 percent of this group participated.

Voters chose a three-member presidency for Bosnia-Herzegovina that will include one Muslim, one Croat and one Serb. They also were electing a Bosnian parliament, a federation parliament and a Serb national assembly. Additionally, they were choosing presidents for the federation and for the Bosnian Serb republic and 10 cantonal assemblies in the federation.

The presidency of Bosnia-Herzegovina and the national parliament are the key institutions envisaged by the Dayton agreement for establishing a unified state. The other bodies that were being created were designed to give a wide measure of autonomy to the three ethnic groups within that state.