Iraqi officials said that initial indications were that as many as 11 million people cast ballots, which, if the estimate holds, would put the overall turnout at more than 70 percent. With Iraqis still lining up to vote in front of ballot centers as the sun went down, officials ordered the polls to stay open an extra hour.

In the country's Sunni Arab neighborhoods, hundreds of thousands of Iraqis who had boycotted the election in January came out this time. ''Last time, if you voted, you died,'' said Abdul Jabbar Mahdi, a Sunni who brought his wife and three children to a polling station in the ordinarily tense Baghdad neighborhood of Adhamiya. ''God willing, this election will lead to peace.''

The day was peaceful, even in areas normally beset by violence. With more than 375,000 American and Iraqi troops and police spread out across the country, the American command here reported 52 attacks, fewer than usual, with 18 of those against polling centers. In January, when Iraqis elected a transitional government, insurgents attacked nearly 300 times, a third of the time against polling places.

In villages and towns, in the Shiite south and in the Sunni Triangle, Iraqis streamed to the polls.

The day's events seemed a significant triumph for Iraqi officials and for the Bush administration, which has long maintained that the democratic process would begin to draw ordinary Sunnis away from the insurgency and encourage them to support democracy's success.

Iraqi officials said that election results would probably not be available for several days, possibly not even until January.

In Washington, President Bush hailed the election's success. ''This is a major step forward in achieving our objective, which is having a democratic Iraq, a country able to sustain itself and defend itself, a country that will be an ally in the war on terror, and a country which will send such a powerful example to others the region, whether they live in Iran or Syria,'' Bush said.

If the drop in violence lasts beyond election day, it could strengthen the Bush administration's tentative plans for carrying out significant reductions in the number of American troops next year.

The election, carried off by the Iraqis with help from the Americans and the United Nations, was, in a country at war, a logistical wonder. The Iraqis opened 6,048 polling centers, which were attended by about 300,000 election observers. Poll workers tacked up or handed out about 5 million informational posters.

Voters cast ballots to fill 275 seats in the Iraqi Parliament, called the Council of Representatives, with at least a quarter of the seats reserved for women. The Parliament will select a president, and approve the prime minister and his cabinet.

The electoral system now in place should allot the Sunnis representatives in numbers equal to their proportion of the population, which is thought to be between 15 and 20 percent. In the current government, which was chosen under a different electoral system, Sunnis have a tiny number of seats.

No single party is expected to claim an outright majority. The leaders of many of Iraq's political parties say that the main political parties here will soon begin what is likely to be a protracted period of negotiations to form a coalition government.