The vote count is expected to take several days
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World leaders have praised the conduct of Iraq's first multi-party elections for more than 50 years.
President Bush and UK Prime Minister Tony Blair - the leaders of the two nations which led the invasion of Iraq - hailed them as a resounding success.
And UN Secretary-General Kofi Annan said Iraqis should be encouraged to take control of their own future.
After what officials said was a higher turnout than expected, the next few days will be spent counting votes.
Preliminary results are expected in about six days, with a full result not due for 10 days.
But correspondents say there was a marked division in voting - high in Shia and Kurdish strongholds and much lower in Sunni Arab areas.
The election was marred by a series of election-day attacks across Iraq which killed at least 36 people.
But Iraqi Prime Minister Iyad Allawi described the poll as a "victory over terrorism".
'Moving and humbling'
Mr Bush congratulated the Iraqi people on a "great and historical achievement".
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A large number of people have shown up to vote: men and women, Muslims and Christians, Sunnis and Shias
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"The Iraqi people themselves made this election a resounding success," he said.
"They have demonstrated the kind of courage that is always the foundation of self-government."
In London, UK Prime Minister Tony Blair hailed the election as "moving" and "humbling".
The Iraqis "came out despite the dangers," he said.
Countries which opposed the invasion also said the election had been good news for Iraqis.
French President Jacques Chirac described them as a "great success for the international community", while a spokesman for German Chancellor Gerhard Schroeder said the high turnout showed Iraqis wanted to take their future into their own hands.
Mr Annan, meanwhile, said the Iraqis had shown courage.
"The Iraqis who turned out today are courageous, they know that they are voting for the future of their country," he said.
"We must encourage them and support them to take control of their destiny."
'Eight million votes'
One of Iraq's most influential religious figures, the senior Shia leader Grand Ayatollah Ali al-Sistani, thanked Iraqis for voting.
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ELECTION SCHEDULE
Sunday: Polls opened for 10 hours, with an extension for those still queuing to vote at 1700 (1400 GMT)
Vote counting for four or more days
Early March: PM appointed
Late March: Government formed
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He said he regretted not being able to do so himself because he is Iranian by birth.
Electoral officials estimated that up to eight million Iraqis voted - more than 60% of those registered.
Tens of thousands of Iraqi expatriates in 14 other countries also voted in a mainly peaceful atmosphere, although scuffles broke out in the UK.
More than 200 parties and coalitions are competing for seats in the transitional assembly, which will draft a new Iraqi constitution ahead of planned elections for a full-term parliament.
Voting at polling stations in the country's south and north was brisk, and there were smiles and tears of joy among voters.
But reports from central Sunni cities say not all polling stations opened, and many voters stayed away out of fear of attack or opposition to the election itself.
Authorities had imposed an unprecedented series of security measures - including shoot-on-sight curfews, closed foreign borders, a ban on cars and travel restrictions within Iraq.
Despite the measures, the capital was hit by nine suicide bombings and a number of mortar attacks.
In an internet statement, a group said to be led by militant leader Abu Musab al-Zarqawi claimed to be behind some attacks in Baghdad and Mosul.
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