Egyptians are voting in the country's first contested presidential election, which incumbent, Hosni Mubarak, is widely expected to win.
Voters have come in trickles rather than droves, reports say
The 77-year-old leader - who has ruled Egypt for 24 years - is being challenged by nine other candidates.
Voting in the capital, Cairo, was slow and there are reports of some irregularities at polling stations.
A demonstration organised by the opposition Kifaya group was held in the centre of Cairo calling for a boycott.
Hundreds of people gathered in the city's Tahrir Square, chanting slogans like "No to corruption" and "No to Mubarak".
They came despite a government warning that it would not allow demonstrations on election day.
"I've never been in a demonstration before, I've never done anything before. But I'm disgusted. I've had enough," accountant Ahmad Ibrahim al-Shimi told the Associated Press.
Reuters news agency quoted witnesses as saying that men in plain clothes had begun chasing and beating demonstrators, and ripping up banners.
Opposition activists said the men worked for the security forces.
The head of an independent judicial committee monitoring the vote said that so far complaints were limited.
But there have been reports of some problems, such as people being allowed to vote without voting cards and not being marked with indelible ink, or receiving money and goods in return for their votes.
Government supporters crashed the Kifaya rally in Cairo
Rights groups have also complained of intimidation of voters and monitors by officials and supporters of Mr Mubarak's ruling National Democratic Party.
Leading challenger Ayman Nour said abuses undermined the credibility of the vote.
"There are no fair elections taking place," he told a news conference. "We accept only the results of free and fair elections."
A court said on Tuesday that organisers could stop independent groups from monitoring voting at polling stations.
The ruling by Egypt's Supreme Administrative Court overturned last week's decision by a lower court.
International monitors are not being allowed to observe the elections either.
About 10,000 polling stations opened at 0800 local time (0500 GMT) and will close at 2200 (1900 GMT).
On the face of it, Egyptians are spoilt for choice as they can select a new president from 10 candidates, says the BBC's Heba Saleh in Cairo.
If no candidate polls 50%, the two with the highest number of votes proceed to round two
Egyptians over 18 are required to vote by law. There are over 32m registered voters out of a population of some 74.9m
Presidents elected for six year term
All were able to campaign freely and they were given equal time on television, she says, but there is little chance that anyone other than Mr Mubarak will win.
He is the only leader the majority of Egyptians have known and most of his challengers are not even taken seriously as politicians.
Only two can be said to have constituencies.
One is Mr Nour of the Ghad (Tomorrow) party, an outspoken young liberal who says he would rule for two years only during which he would lay down the foundations of a democratic state.
His detention earlier this year earned Egypt public criticism from the US.
The other serious contender, Nomaan Gomaa, heads the Wafd, the once vigorous liberal party that fought for independence from Britain but that has been seriously weakened by the political restrictions of the past 50 years.
The biggest opposition group, the Muslim Brotherhood, is excluded from the election.