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Philippines election: Polls open with outspoken mayor tipped to win
President Benigno "Noynoy" Aquino is standing down as the constitution limits presidents to one six-year term. Filipinos will also pick a vice-president and local officials.
The election campaign has focused on reforming the economy, infrastructure, tackling corruption and crime and on the territorial disputes with China in the South China Sea.
Mr Duterte has run a campaign focused on law and order issues, but made many controversial statements, including saying that he would butcher criminals.
A former state prosecutor nicknamed "The Punisher", he has been mayor of the southern city of Davao for more than 22 years.
He recently joked that, as mayor, he should have been first to rape an Australian missionary murdered in a prison riot, but he later apologised.
At the scene: Jonathan Head, BBC South East Asia correspondent
Elections here are a cheerful, communal affair with large family groups or neighbours walking into the polling stations to vote together.
The constant crowds coming in and out of the elementary school in Manila's Tondo district looked hard to manage. But election officials are well-practised, and voters well-informed.
Plenty of officials were on hand to help voters manage the formidable ballot sheets, listing in this constituency dozens of candidates for the various local and national posts.
The enthusiasm across the age and class spectrum for Mr Duterte in this campaign has exposed the weariness of Filipinos with the familiar political faces, who have delivered some economic improvements but little real change in the levels of poverty and corruption.
Mr Duterte has suggested he will disregard democratic checks and balances if they get in the way of fixing the country's problems.
It's a message that has excited and attracted people. Yet the numbers coming to vote here, and the positive and relaxed atmosphere, show that the faith in the familiar rituals of democracy is still as strong here as anywhere in Asia.
Mr Duterte's closest rivals in the opinion polls are Grace Poe - a former schoolteacher and first-term senator - and Mar Roxas, a former investment banker and the grandson of the first president of the Philippine Republic.
President Benigno Aquino has been leading attempts to bring together other candidates in an effort to defeat Mr Duterte.
He warned that if Mr Duterte were to be elected, it could mean a return to dictatorship.
In a final rally on Saturday, Mr Aquino appealed to voters: "I need your help to stop the return of terror in our land. I cannot do it alone."
However the other four candidates - also including Jejomar Binay and Miriam Defensor-Santiago - refused to step aside.