Among Iraqis living outside their homeland, about 20,000 registered to vote in Jordan. Election officials there reported a high turnout.
Election officials in Jordan say they are happy with the turnout. The Geneva-based International Organization for Migration organized the voting for Iraqis living abroad.
Lazhar Aloui, who heads the IOM mission in Jordan, says he is pleased with the turnout and with the Iraqis who supported the process.
"The idea is really not to make sure that every Iraqi everywhere outside Iraqi will vote, but its really to provide an opportunity for most of the people to participate in this first step in the democratic transition," he says.
Mr. Aloui acknowledges that there are many thousands of Iraqis in Jordan who are not voting, however.
Even though over 20,000 Iraqi expatriates registered to vote in Jordan, that represents only a small percentage of the Iraqi community in the country. Jordanian officials say there are more than 200,000 Iraqis in Jordan, but other estimates place the number far higher, approaching half a million.
Eleven polling stations were set up in several cities in Jordan, eight of them here in in Amman.
Outside the Al-Hussein voting center in central Amman, 24-year-old Ahmed Al-Jobouri says voting in the landmark election was too important to pass up.
Mr. Al-Jobouri says his vote is a message for friends and family back in Iraq.
"The Iraqi man didn't know what freedom means, so we helped here to let people know what's the meaning of freedom," Mr. Mr. Al-Jobouri says.
But, reports out of Iraq itself were cause for concern, and many of the voters in the Al-Hussein center traded news of suicide bomb attacks in Iraq.
Thirty-year-old Fadia says her family in Baghdad heard explosions, as they prepared to vote Sunday morning.
Fadia prefers to give only her first name, and says that, if she were in Iraq, she might be too scared to vote. But here in Jordan, she says, she was able to take part.
"It feels free. Having a chance to make any choice you want - something like Iraqis never had before. And I am one of them," Ms.Fadia says.
More than 280,000 Iraqi expatriates registered to vote in 14 countries, including Jordan, Syria, Iran, Britain, Sweden and the United States.
Iraqis are voting for 275 members of a national assembly, which will in turn draft and approve a new constitution. They are also choosing members of 18 provincial assemblies and an autonomous Kurdish parliament in northern Iraq.
Results of voting in Jordan are expected within a few days.