(Transcript from World News Australia Radio)
The vote is being seen as a prelude to a presidential bid by Egypt's army chief, General Abdel Fattah al-Sisi, in elections due to follow later this year.
This is the first vote since the military-backed government took over in July after overthrowing Egypt's first democratically elected president, Mohamed Morsi of the Muslim Brotherhood.
(Click on audio tab above to hear full item)
The new charter has done away with much of the Islamist-inspired wording of Mohamed Morsi's constitution, which was suspended on his overthrow.
Its supporters say it expands women's rights and freedom of speech.
Human rights group Amnesty International claims the new draft constitution is an improvement on the 2012 constitution but says it still falls short of international human rights standards.
Amnesty claims the draft allows for military trial of civilians, discriminates against foreign nationals and leaves room for undue restrictions to be placed on freedom of expression and assembly.
Shortly before polls opened, a small bomb exploded outside a Cairo court.
No fatalities or injuries have been reported.
It came as hundreds of thousands of soldiers and police were deployed to guard polling stations.
Polling at most stations got off to a smooth start, but local officials say five people have been killed in central and southern Egypt in clashes between police and supporters of ousted president Mohamed Morsi.
It is unclear how many Egyptians will vote because of security worries, but the constitution appears certain to pass.
Eighty observers from Democracy International were deployed in 23 governorates across Egypt.
Democracy International's Program Manager, Dan Murphy, says teams have been monitoring the situation in the weeks leading up to the vote.
"We have been on the ground in Cairo since the beginning of December and we have met with a variety of stakeholders, political parties, civil society groups, government entities. We've talked to everyone to try and get a very clear picture of what's going on. When we make a statement it will include our observations on that process as well."
Dan Murphy says the vote is off to a good start.
"So far our teams have been provided good access and we've made lots of observations and we think that the process from a technical standpoint today is proceeding relatively well."
An Islamist coalition led by the former president's Muslim Brotherhood had encouraged protests and a boycott of the two-day vote.
But many Egyptians are showing their support for the vote.
Resident Nabil Matta says he has faith in the new constitution.
"I came out because this constitution is the best constitution we have had and was done by professionals. First of all it respects human rights and freedom of religion. It's a civil constitution for a civil advanced state and this is what we were wishing for."
This Egyptian woman, Malaka Mohamed, says she hopes the vote may bring peace and stability to the country.
"Yes to the constitution and may God bring calm to the country. Many people have been lost, and may God bring us victory."
The United States has voiced concerns about outbreaks of violence during Egypt's referendum, saying it won't help the country's political transition.
State Department deputy spokeswoman Marie Harf says regardless of the vote's outcome, it must pave the way for a peaceful future.
"Regardless of whether the constitution is approved in the referendum, it will be important for the interim government to foster a positive environment for civil society and to protect the rights of political activists and groups to peacefully express their views on the country's future."
But Ms Harf stressed that voting is not yet complete, and the US government awaits reports from Egyptian and international independent observers.
Results from the referendum are expected within three days of the polls' close.
