Ukraine is in the presidential polls after months of unrest following the ousting of former President Viktor Yanukovych.
Eighteen candidates are competing in the contest, which is widely seen as a crucial moment to unite the country.
But pro-Russian separatists in the east who oppose the election have threatened to disrupt the voting process.
Some 20 people have been killed amid an upsurge of fighting between insurgents and government forces in recent days.
The violence in the east, particularly Donetsk and Luhansk, has seriously disrupted preparations for the polls.
Seven out of 12 district election commissions have opened across the region of Donetsk, and none in the cities of Donetsk or Horlivka.
Two district commissions are open in Luhansk.
The presidential elections were called after the last elected President, Viktor Yanukovych, was deposed in February amid mass protests against his pro-Russian policies.
Confectionary tycoon Petro Poroshenko, known as the “chocolate king”, is the favorite to win.
Former PM Yulia Tymoshenko is lagging behind Petro Poroshenko in opinion polls.
In order to win outright, one candidate must receive more than 50% of the vote, otherwise a second round of voting will be held on June 15.
Interim PM Arseniy Yatsenyuk urged people to vote and “defend Ukraine” in a televised address on Saturday.
“This will be the expression of the will of Ukrainians from the west, east, north and south,” he said.
In an unprecedented move, Russian President Vladimir Putin said on Friday that he would respect the outcome of the election and was prepared to work with whoever was elected president.
It comes after months of tension with Russia, which has been blamed by Kiev and its Western allies of stoking separatist sentiment in eastern parts of Ukraine. Vladimir Putin has denied the claims.
Over 1,000 observers for the Organization for Security and Cooperation in Europe (OSCE) have been deployed nationwide at polling stations.
But the OSCE has pulled out most of its observers from the eastern Donetsk region over fears for their security.
Some pro-Russian separatists have warned people against voting, with reports of election officials and voter lists being been seized at gunpoint.
“If necessary we will revert to the use of force,” Denis Pushilin, a leader of the self-declared Donetsk People’s Republic, is quoted as saying by the AFP news agency.
Separatists in Donetsk and Luhansk declared independence from Ukraine after holding referendums on May 11, a move not recognized by Kiev nor its Western allies.
The two regions took their cue after a disputed referendum in Crimea, which was annexed by Russia in March.
More than 75,000 police and volunteers are said to have mobilized to ensure security during the vote.
Polling stations will remain open until 20:00, with definitive results expected on Monday.
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