Amid intense security, including a ban on all traffic on Saturday, voters walked to polling stations in Baghdad as they opened at 7am (0400 GMT).
Iraqi journalist Walid Khalid told Aljazeera from Baghdad that the voting process was going calmly though slowly.
However, there were little signs of voting in the Baghdad districts of al-Ghazaliya, al-Mansur, al-Amiriya and Dora.
In the southern city of Basra and the northern oil-rich city of Kirkuk, voter participation was reported to be strong.
Falluja, which was the site of heavy fighting between US forces and Iraqi fighters in 2004, also reported a large voter turnout.
However, the town of Ramadi west of Baghdad, and the northern city of Mosul reported very little voting.
Iraqi journalist Muhammad Abd al-Rahman told Aljazeera that in Hiyt, in western al-Anbar, some residents did not know where the polling centres were.
Iraqis’ fingers are marked with
In Mosul, Iraq’s third largest city, threatening fliers urged people to keep away.
The fliers showed an American Uncle Sam figure looming over a voter depicted as a donkey placing a ballot paper into a voting box that is shown at the bottom to be a meat shredder.
“Stay home; don’t believe in the constitution,” the caption read.
More than 100,000 Iraqi police and soldiers will protect the more than 6000 polling stations over the day, with US and other foreign troops on standby.
But the tight security was not enough to stop a roadside bomb from exploding on al-Madhif street in al-Amiriya area, targeting an Iraqi police patrol stationed near a polling centre.
Two police officers were seriously injured and a police vehicle was damaged, Khalid told Aljazeera.
Attackers have struck early
The explosion occurred as the centre was opening at 7am, and no voters were there, said police Lieutenant Muhammad Kheyon.
Fighters also attacked a polling centre in al-Dhuluiya, north of Baghdad, clashing with US forces who were backed by helicopters.
A civilian was injured when a rocket-propelled grenade hit a polling centre in al-Adhamiya.
Police sources told Khalid that explosive devices were planted along roads leading to polling centres in al-Amiriya, and bomb experts had been called in.
In Baquba, three Iraqi soldiers guarding a polling centre were killed and three others injured in a bomb explosion.
In the city of Ramadi, fighting erupted at about 7am between a small group of fighters and US and Iraqi troops patrolling the mostly empty streets of the city, said police 1st Lieutenant Muhammad al-Ubaidi.
Mortar and rocket blasts and machine-gun fire echoed around the regional governor’s compound in the city centre.
Clashes could also be heard in other parts of Ramadi, the capital of al-Anbar province.
It was not immediately clear whether anyone was wounded or killed.
Aljazeera learned that Ramadi residents had stayed at home frightened by the presence of US forces. The heavy gunfire heard in the morning had scared voters, he said.
The journalist added that US warplanes were hovering over Ramadi city at low altitude.
South of Basra, three armed men who attacked an empty polling station at 3am (2300 GMT) were caught and arrested, said police Captain Mushtaq Kadhim.
On Friday night, four polling stations in southern Baghdad were fired upon, an Interior Ministry source said. No casualties were reported.
The attacks may cause delays that may push back the polling centres’ scheduled closing time of 5pm. Up to 15 million Iraqis are set to accept or reject the new constitution.
Al-Qaida in Iraq has condemned the vote on the first post-Saddam Hussein charter. The group’s leader, Abu Musab al-Zarqawi, has called on the minority Sunni Arabs to boycott.
Despite the violence and some polling centres in al-Anbar province and the Mosul area being shut, Iraq’s Electoral Commission said things were going well overall, with the vast majority of the polling stations open.
Carina Perelli, who leads a UN mission overseeing the election, also said that the voting was going well.
In Baghdad’s fortified Green Zone compound, where the Iraqi government headquarters is, President Jalal Talabani and Prime Minister Ibrahim al-Jaafari were two of the first to cast votes.
The constitution is backed by Shia
“I voted ‘yes’ and I urge all Iraqis, no matter their different ethnicities and religions … to vote ‘yes’ to the constitution,” he said.
Talabani, a Kurd, said the choice was between political action and violence.
“The Sunni Arab brothers should understand that their aspirations will be achieved through political action, and not violence and terrorist acts,” he said on Friday night on a television station popular with Sunnis.
Many in the Sunni community, as well as some in the Shia community, fear that federalist provisions in the document will lead to the country splitting into three parts – with a Shia south and a Kurdish north holding the oil wealth and the Sunnis left empty-handed in the centre.
The Kurds overwhelmingly favour the document, as do the Shia.
The draft needs a simple majority of the 15.5 million registered voters for approval. But it can be blocked if two-thirds of voters in any three of the 18 provinces vote against it.
Sunni Arabs have a majority in al-Anbar, Salah al-Din, Nineveh and Diyala provinces.
But the Sunnis are split, particularly since Shia and Kurdish legislators reached a deal with Sunni colleagues on Wednesday approving last-minute additions to the draft.
These include the creation of a panel to consider amendments after general elections on 15 December.
Since then, six offices of the Islamic Party, which has come out in favour of the draft, have come under attack.
And a group with ties to al-Qaida issued a death threat via the internet against Islamic Party leaders.