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Ukraine's President Petro Poroshenko has called elections being held in the rebel-held east of the country illegitimate.
He said that Ukraine and the rest of the world would reject the results, and called on Russia to do the same.
Pro-Russian separatists will vote to set up a breakaway regional leadership in the eastern Ukraine region of Donetsk and Luhansk aiming to take their war-torn region closer to Russia and defying Kiev and the West as the big guns still boom across the territory.
Ukrainians vote in an election that is likely to install a pro-Western parliament and strengthen President Petro Poroshenko's mandate to end separatist conflict in the east, but could fuel tension with Russia.
People wrapped up warmly on a cold, clear day to vote in the first parliamentary poll since protests in the capital Kiev last winter forced Moscow-backed leader Viktor Yanukovich to flee and ushered in a pro-Europe leadership under Poroshenko.
In eastern regions controlled by the army, soldiers armed with automatic rifles and wearing bulletproof jackets guarded polling stations under the yellow and blue Ukrainian flag.
At least 10 people have reportedly been killed after shelling of a school playground and a nearby street in the Ukrainian city of Donetsk.
Witnesses quoted by Reuters said a biology teacher and a parent of one of the school's pupils were killed during the blasts, although no children were among the dead.
The area has been a stronghold of pro-Russian separatists since the beginning of the conflict in eastern Ukraine.
Protesters have pulled down one of the biggest statues of communist leader Vladimir Lenin in Ukraine.
Hundreds of nationalists cheered and danced around as the monument came crashing down in the country's second-largest city, Kharkiv.
Lenin is considered by many Ukrainians as an undesired link to Moscow.
The tearing down of the memorial is said to be supported by Kiev officials.
After it fell most people in the crowd rushed to the ruins to collect a small piece of the once powerful symbol to take home.
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Russian Foreign Minister Sergei Lavrov was quoted on Sunday as saying that the time had come for his country to modernise its nuclear and conventional arsenals.
"I don't think we are on the verge of a new arms race. At least, Russia definitely won't be part of it. In our case, it's just that the time has come for us to modernise our nuclear and conventional arsenals," Lavrov said, according to a transcript of an interview with Russia Today television published on its website.
The ceasefire in Ukraine has seen its first full day without any reports of deaths or casualties, the country's President has said.
Petro Poroshenko called the development a sign that the truce has "finally begun working".
The world must not lift sanctions against Russia until Ukraine regains control of its entire territory, the country's prime minster has said.
Ending a day of intense discussions over the growing global threat of extremist groups, Arseniy Yatsenyuk told the United Nations General Assembly of world leaders that "we know what terrorism means".
He demanded that Russia pull back its forces from eastern Ukraine, "stop the supply of Russian-led terrorists" and start "real talks, peace talks".
A "significant" number of Russian troops have now withdrawn from Ukraine, a Nato spokesman said.
Lieutenant-Colonel Jay Janzen told Reuters: "There has been a significant pullback of Russian conventional forces from inside Ukraine, but many thousands are still deployed in the vicinity of the border.
"It is difficult to determine the number, as pro-Russian separatists control several border crossings and troops are routinely moving back and forth.
"Further, Russian special forces are operating in Ukraine, and they are difficult to detect."
Latest ITV News reports
Ukraine's curiously named Internet Party has come up with an offbeat campaign gimmick to lure voters - dressing as Star Wars characters.
Tensions remain high in eastern Ukraine after at least three died in a day of shooting during what was supposed to be an Easter truce.