Polls closed at 6 p.m. on Saturday - an hour later than planned - after millions of voters cast ballots for influential regional councils around most of Iraq. There were no reports of major violence.
Iraqi authorities imposed a huge security operation around the country that included traffic bans in major cities and extensive checkpoints and surveillance posts.
The U.S. military also was out in force but did not take a direct role in the election security.
Results from the elections are not expected before Tuesday.
In Washington, President Barack Obama congratulated Iraqis on Saturday's provincial elections and urged the winners to work for their people.
President Obama said the elections for the provincial council seats were significant, peaceful and important steps toward Iraqis taking responsibility for their future. His statement did not address complaints from local officials that the Shiite-led government attempted to keep the minority Sunnis from full participation.
"Millions of Iraqi citizens from every ethnic and religious group went peacefully to the polls across the country to choose new provincial councils," Mr. Obama said in a statement released by the White House. "It is important that the councils get seated, select new governors and begin work on behalf of the Iraqi people who elected them."
"The Purple Fingers Have Come Back To Build Iraq Again"
Even before a single ballot was counted, Iraqi officials were basking in the successes - watching millions of voters wave the purple-tinted fingers that have become symbols of the country's hopes for a workable democracy.
But election observers and others were examining a growing list of complaints, including claims that hundreds of people - perhaps more - were wrongly omitted from voting lists in areas across Iraq.
"There was huge amount of confusion," said Afram Yakoub, a Belgium-based election monitor who visited polling sites in the Mosul area in northern Iraq. "Names were on the center voter registry but did not appear on the (polling) station registry."
The leader of the second largest Sunni bloc in parliament, Saleh al-Mutlaq, accused the Shiite-led government of a deliberate campaign to keep the minority Sunnis "on the sidelines."
It was unclear whether the alleged problems were isolated or could cast doubts on the entire election.
But any political bitterness could further complicate another difficult task ahead for Iraq's leaders: getting hundreds of factions to accept the results as credible and then start hammering out alliances from among 14,000 candidates for the influential regional posts.
The overall picture, however, was close to the goals set by Iraqi officials desperate to portray a sense of order and confidence nearly six years after the U.S.-led invasion.
A vast security operation faced no major confrontations or attacks. Meanwhile, Sunni groups - that boycotted provincial elections four years ago - were deeply involved in the election.
"The purple fingers have come back to build Iraq again," said Prime Minister Nouri al-Maliki in a nationwide address shortly after the polls closed - referring to the ink used to identify those who cast ballots.
Initial results are not expected for several days. But possible challenges were already leaking out.
A senior Sunni leader in the western Anbar province - where former anti-insurgent militias were seeking political gains - alleged that voters couldn't reach polling stations because of the traffic ban and others in Fallujah found the doors shut.
"We expect fraud ... Some will try to fill these blank ballots," said Sheik Dari al-Arsan. "We will complain about these violations."
In the southern Shiite city of Basra, voter Hadi Thegil stared angrily at election workers when he was told he wasn't on the registration list, which is compiled using information form Iraq's ration card system. He left muttering: "I feel robbed."
In Karmah, about 50 miles west of Baghdad, local election observer Sabah Hussein said he found ballots marked in advance for the Iraqi Islamic Party, a Sunni bloc that's a partner in al-Maliki's government. It was unclear whether any others were cast.
But a U.N. election observer, Said Arikat, described the election in mostly positive terms.
"By and large, the rules were followed. We weren't aware of any confusion in the stations we visited. I am sure there will be complaints, and I'm not sure you can guard against a total absence of such complaints."
A Shiite lawmaker, Nassir al-Saadi, also found the election process generally good but noted the real test is yet to come: how the major political bloc perceive the outcome.
"The only real gauge whether the election is credible or not is the results," he said. "If the results are fair then we can say the election was fair."
It will be a huge job sorting it all out. A total of 440 seats are at stake on the various provincial councils in the election - covering the whole country except four northern areas.
The winners then will have to forge working coalitions from a potential patchwork: veteran political groups amid the many newcomer candidates. There also are still questions about how to ensure sufficient representation from the approximately 3,900 women candidates.
Turnout figures were not immediately available for the 15 million eligible voters. Election workers at various sites around the country reported steady streams of voters but few huge crowds, and voting was extended for one hour.
The voting began under a security net that appeared even more extensive than Iraq's last elections in 2005. Voters passed through several choke points and then individually searched - men in the open by police and women in tents by teams that included female teachers and civil workers.
In some parts of Baghdad, checkpoints were spaced 30 yards apart and Iraqi security forces, including special forces in combat gear, conducted foot patrols.
U.S. soldiers were also out in force, but remained well away from polling centers. The U.S. military assisted in security preparations for the elections, but said troops had a backseat role in the election day operations.
There were reports of isolated violence and unrest.
In Tikrit, Saddam Hussein's hometown about 80 miles north of Baghdad, three mortar shells exploded near a polling station, but caused no casualties, said police, who spoke on condition of anonymity because they were not authorized to speak to media.
Hundreds of Iraqi Kurds stormed an election office in the disputed northern city of Khanaqin after claiming many of them were not on voting lists. There were no reports of serious injuries. The incident was part of lingering disputes between Kurds and the Arab-run central government over control of the city near the Iranian border.
Each region carried its own distinctive mood.
In the Kurdish autonomous region - which is scheduled to hold elections later - special polling sites were created for Iraqis who have sought refuge from violence in other parts of Iraq. "I hope the real winner will be Iraq itself," said Mohammad Rasid, 75, who fled Baghdad two years ago.
In nearby Mosul, considered one of the last urban strongholds of al Qaeda in Iraq and other insurgent groups, Sunni Arab parties urged for a high turnout to counter Kurdish ambitions to extend their influence over the city.
"I came to take back my city for Sunnis," said Afifa Abdul-Nafaa, 81, who came to vote in a wheelchair pushed by her son.
In the western Anbar province, the Sunni tribes that rose up against al Qaeda and other insurgents - and led to a turning point of the war - are now seeking to transform their fame into council seats and significantly increase their role in wider Iraqi affairs. Turnout in Anbar was about 2 percent in provincial elections four years ago.
And in Iraq's Shiite south, loyalists to prime minister al-Maliki appeared to receive a boost from the offensives last year that broke the hold of Shiite militias in the key city of Basra and other places.
"When the militiamen were in charge, we used to see bodies laying in the street," said Ali Majid, 25. "Now we have some order."
Zakiya Tahir, a 71-year-old woman who cannot read, pointed to a poster of a local candidate supported by al-Maliki.
"I have nothing to do with politics," she said. "I just want to feel safe again."
2 Iraqi Police Killed After Firing On U.S. Troops
The U.S. military says two Iraqi policemen have been killed after they opened fire on U.S. soldiers conducting an operation against al Qaeda near the northern city of Mosul.
A U.S. statement says the incident began when U.S. soldiers entered a neighborhood looking for a suspected al Qaeda operative who was believed to be in one building.
The statement says the soldiers began taking fire from another building and after identifying themselves fired back.
U.S. soldiers and Iraqi police entered the house and found two Iraqi policemen in civilian clothes had been killed in the exchange.
A U.S. spokesman says coalition forces "deeply regret" the loss of the two policemen.