The executive director of the Iraqi Electoral Commission, Ali al-Lami, appealed for peace on Thursday, when about 15 million people will be called on to vote in more than 6,200 polling stations.
Insurgents have denounced the election as a "satanic project" but have not threatened to attack polling stations.
U.S. and Iraqi forces are determined violence won't be able to stop the election Thursday. The country is virtually locked down, and raids against insurgents are ongoing, reports CBS News correspondent Lara Logan.
In other developments:
Voting abroad began first in Australia, where up to 20,000 registered Iraqi voters live. They are part of a group of 1.5 million voters living outside Iraq who will cast ballots at polling centers in 15 countries, including the United States and Canada.
An estimated 240,000 Iraqi-Americans are believed eligible to cast their votes. Sandra McNeill of CBS radio station WWJ-AM reports there are two election sites in Detroit, and an organizer expects at least twice as many voters as last year, when bad weather and poor choice of sites kept many voters away.
Eligible voters can be American citizens. But they must be at least 18-years-old, have been born in Iraq and hold citizenship there. Iraqis born in the United States who can prove their father is Iraqi also can vote.
The expatriates will help elect the 275-member National Assembly, which will legislate in the coming four years and choose the first fully constitutional government in Iraq since the collapse of Saddam's rule in 2003.
Low turnout was recorded at 12 ballot centers across Jordan.
"Why should I vote?" asked Umm Zainab, who sat on a street corner in Amman's bustling downtown.
"All the candidates are bad and they're not going to bring us security and prosperity," added a 42-year-old cigarette vendor from the Shitte holy city of Karbala. She declined to give her name in line with conservative Islamic tradition.
In the Jordanian city of Zarqa, Hamed Al-Nasseri, 56, said he was voting to rid his nation of militants. "What they say is bogus. I'm voting to dare these militants, to have a strong parliament and government that would restrain these outlaws," he said.
Early voting was held Monday for Iraqi security forces, hospital patients and prisoners, and proceeded without problems, al-Lami said.
Gunmen in the insurgent stronghold of Ramadi, west of Baghdad, killed Sunni Arab candidate Mezher al-Dulaimi while he was filling up his car at a gas station.
A roadside bomb targeted the convoy of Sheik Jalal Eddin al-Sagheer, a Shiite member of the National Assembly who was elected with the governing United Iraqi Alliance. The Iraqi army said the explosion in Latifiyah, about 20 miles south of Baghdad, damaged one of the vehicles.
Deputy Prime Minister Ahmad Chalabi held a pre-election rally in the southern city of Basra for about 1,000 supporters, while former Prime Minister Ayad Allawi hosted a gathering in Baghdad. No campaigning is allowed Wednesday to give Iraqis time to reflect ahead of the election.
Abdul-Aziz al-Hakim, leader of the Supreme Council for the Islamic Revolution in Iraq, told about 1,000 tribal leaders who gathered in Baghdad's Jadriyah neighborhood that the military wing of his group — the Badr Brigade — was ready to help with election security.
"I declare that the Badr Organization is ready to mobilize 200,000 of its men in all parts of Iraq so that they can play a role in defending Iraqi and Iraqis," said the black-turbaned cleric, who is heading the strong Shiite United Iraqi Alliance slate.
"Violence has no place in any democratic elections. This is a time for national reconciliation though the political process," al-Lami said.
In a rare joint statement, Al Qaeda in Iraq and four other Islamic extremist groups denounced the election as a "satanic project" and said that "to engage in the so-called political process" violates "the legitimate policy approved by God."
The groups vowed to "continue our jihad (holy war) ... to establish an Islamic state ruled by the book (the Quran) and the traditions of the Prophet Muhammad."
However, the statement contained no clear threat to disrupt voting, unlike the Jan. 30 election for an interim parliament and the Oct. 15 referendum on the constitution.
The authenticity of the statement could not be verified, but it appeared on a Web site that often publishes extremist material.
The absence of a clear-cut threat could reflect the growing interest among Sunni Arabs, the foundation of the insurgency, to take part in the election. The Sunni decision to boycott the January ballot left parliament in the hands of Shiites and Kurds — a move which increased communal friction and cost the Sunnis considerable influence in drafting the constitution.
A leaflet that appeared Monday in the Baghdad Sunni stronghold of Azamiyah acknowledged that Sunni Arabs could make gains in the election but that "fighting will continue with the infidels and their followers."
The statement was unsigned but was written in a style favored by Islamic extremists.
Americans hope a successful election will enable them to start pulling out U.S. troops next year, something many Iraqi people desperately want to see, reports Logan.