Results are expected on 5 June, and there are few doubts that the former defence minister Abdul Fattah al-Sisi will win comfortably.
The only other candidate is veteran left-winger and former presidential contender Hamdeen Sabahi.
Why is the election being held early?
The army ousted Islamist President Mohammed Morsi after mass protests last July, and went on to outline a "roadmap" to democracy that provided for presidential elections two years ahead of schedule.
A parliamentary poll was meant to come first, but the interim government reversed the order in January.
Parliament will be elected later this year, once the president is firmly in power.
The new constitution of January 2014 limits the president to two consecutive four-year terms.
Mohammed Morsi is now on trial accused of inciting the murder of anti-government protesters, murdering prison officers during a 2011 escape from jail, espionage and conspiracy to commit terrorist acts.
What are the candidates' programmes and support base?
Mr Sisi has set out an ambitious plan to develop agriculture, housing, education, impoverished areas and employment.
He is vague on how the massive funds required would be raised, apart from calling for hard work from all and lower profit margins for state and private companies.
He campaigns under the slogan "Long live Egypt!" and says his win would mean the Muslim Brotherhood is "finished" as a political force.
Mr Sisi enjoys the support of several big businessmen, as well as a broad range of political parties from the Islamist right to the moderate left.
Hamdeen Sabahi, a leading left-winger since the 1970s, offers an alternative to young voters who favour a civilian candidate over a military one.
Under the slogan "We will pursue our dream", he promises to combat corruption and incompetence while promoting civil rights.
Like Mr Sisi, however, he says he will not re-legalise Mr Morsi's Muslim Brotherhood, which was banned as a terrorist group.
The parties and movements that back Mr Sabahi are overwhelmingly left-wing.
Abdul Fattah al-Sisi: New face of Egypt's old guard
Who is Abdul Fattah al-Sisi?
Who is Hamdeen Sabahi?
Egyptian election: Voters' views
Is the vote likely to be free and fair?
Two factors work against this. First, the election comes amid security and political turmoil.
The Muslim Brotherhood is continuing to protest against the removal of President Morsi, and the authorities have responded with a violent crackdown on the Brotherhood and its supporters.
Hundreds have died in clashes with security forces since last summer and hundreds of others face jail or even death sentences, prompting further protests.
Second is the overwhelming support Mr Sisi receives from the mainstream media, which are already portraying him as the next occupant of the presidential palace.
Furthermore, the Muslim Brotherhood's own media within Egypt have been banned, so this major opposition voice is restricted to online television and news portals hosted abroad. These do not have anything near the reach of the mainstream media.
Who is monitoring the polls?
Major Western monitoring groups are not observing the vote.
The US Carter Centre says it will not take part, though the European Union says it will monitor the poll across the country.
The Arab League will send 100 monitors, and the African Union will also send a mission despite having suspended Egypt from membership last year.
Who is calling for a boycott?
Several but not all Islamist parties have announced that they will boycott the vote, though some have endorsed Mr Sisi.
The Muslim Brotherhood regards Mr Morsi as the legitimate president and refuses to recognise the new election.
The pro-Morsi National Alliance to Support Legitimacy describes the vote as a "farce", and the Salafist Front has called it "illegitimate".
The political arm of the Gamaa Islamiya (Islamic Group), the Building and Development Party, says most of its members also favour a boycott.
The secular 6 April Youth Movement - prominent in the uprising against Hosni Mubarak - has also called for a boycott, saying the elections were just a way of "enthroning" Mr Sisi.
What problems will the new president face?
At home, the new president will have to tackle severe political, economic, security and social problems.
The public expects the president to deal with unemployment, inequality and corruption, but restoring security may be his single greatest challenge.
Abroad, the new leader will seek to maintain friendly ties with the US and the West, while trying to improve relations with those countries that supported Mr Morsi - Turkey and Qatar and, to some extent, Iran.
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Egyptians go to the polls on 26 and 27 May to elect a new president.