The frontrunner in the campaign has been Nicolas Maduro, Chavez's longtime lieutenant and foreign minister. Maduro, 50, was named acting president after Chavez died of cancer on March 5.
Maduro faces opposition leader Henrique Capriles in the election. Capriles is a 40-year-old state governor who lost to Chavez in October's presidential election by a nearly 11-point margin, the best showing ever by a challenger to the longtime president.
Polls closed at 6:30 p.m. ET and official results are expected later Sunday night.
On Saturday, Capriles repeated his accusation that Maduro broke election rules when he appeared on state-run televisioin during a visit to Chavez's tomb at a military museum in Caracas, alongside Argentinian soccer star Diego Maradona.
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Capriles criticized state television channel VTV on his Twitter feed and said the Saturday broadcast was a "flagrant violation" of a law that prohibits campaigning the day before the election.
Although Maduro is the frontrunner in the election, recent polls said the gap between the two candidates has been narrowing.
Capriles has been pointing out what he calls the "incompetence of the state" in a country that claims to have the world's largest oil reserves and has questioned how the government managed $1 trillion in oil revenues generated in the 14 years Chavez ran the country.
There is frustration among citizens over crumbling infrastructure, unfinished public works projects, double-digit inflation, food and medicine shortages and rampant crime that has given Venezuela among the world's highest homicide and kidnapping rates.
At his campaign rallies, Capriles would read out a list of unfinished road, bridge and rail projects. Then he asked people what goods were scarce on store shelves.
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The opposition contends Chavez used money from the treasury last year to fund government largesse and buy re-election.
In his campaign, Capriles has emphasized the need for reform, says CBC reporter Paul Hunter.
"Capriles has been saying all along that is what Venezuela needs because all those problems are serious problems," said Hunter.
"The problem of course is the support for Maduro is the same crowd that supported Chavez, and that is the country's poor, and they far outnumber the country's wealthy, who typically would support the opposition," he added.
Maduro a closet reformer?
Maduro, a former union activist and bus driver with close ties to Cuba's leaders, constantly alleged that Capriles was conspiring with U.S. putschists to destabilize Venezuela and even suggested Washington had somehow infected Chavez with the cancer that killed him.
But mainly he focused his campaign message on the simple theme of his mentor's October campaign: "I am Chavez. We are all Chavez."
Maduro promised to expand anti-poverty programs, but without explaining how he'd pay for them.
If Maduro wins the election, many believe that the Venezuelan government will remain the same as under Chavez. But some think that Maduro may be a 'closet reformer,'"says Hunter.
"He has hinted that he would like to improve the production in the oil industry," Hunter said. "He has hinted that, while he would maintain ties with Iran and Cuba, maybe it's time to begin a dialogue with the hated Americans and to re-open some sort of relationship with the United States."Suggest a correction