There has been a high turnout at voting stations, reports say.
Opinion polls have put the governing Chama Cha Mapinduzi (CCM) party ahead, but the result is expected to be close.
Four opposition groups are backing one candidate, a former prime minister, in the presidential race.
The BBC's Tulanana Bohela reports from the main city Dar es Salaam that usually busy streets and markets are deserted, and large queues have formed at polling stations as people wait patiently to cast their ballots.
President Jakaya Kikwete, who is standing down after two terms, has called for peace ahead of the election, adding that "anyone who tries to cause trouble will be dealt with".
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Turnout seems to be higher than previous polls, which were a foregone conclusion
There are also long queues in Zanzibar, where previous elections have turned violent
The key issues are said to be clean water, healthcare and educationElection in pictures
CCM was formed in 1977 from a merger of two post-colonial parties and has effectively been in power since independence in 1961.
It has fielded Works Minister John Magufuli, 55, as its presidential candidate.He is being challenged by Edward Lowassa, 62, who quit CCM after he failed to win its presidential nomination.
He is contesting the poll under the banner of the Ukawa coalition.
Analysis: Tulanana Bohela, BBC Africa, Dar es Salaam
The unusually high turnout across the country suggests that Tanzanians believe the future is in their hands.
This is in contrast to previous elections when CCM was certain of victory because of a weak and divided opposition.
But politics has been shaken up by the formation of the Ukawa coalition, which believes it has a real chance of winning.
The CCM is equally confident, setting the stage for the most competitive election since independence from British rule.
Tanzanians are hoping that by Monday final results will be out, and they will know who their next president will be.
Both candidates have already cast their ballots.
"I've carried out my duty as a citizen by voting. My appeal to those Tanzanians who have not voted is to ensure they do so to elect the people they want," Mr Magufuli said.
Mr Lowassa said he was confident of victory, and urged people to vote peacefully.
Last week, he told the BBC Swahili service that he will "go back to his village to rear his cattle" if he loses.
ZANZIBAR.
The semi-autonomous island archipelago of Zanzibar is also voting for a president and local leaders.
It has been hit by violence in previous elections, unlike the mainland where elections tend to be peaceful.
The BBC's Sammy Awami in Zanzibar says that so far, there have been no reports of violence on the islands, with the voting process proceeding smoothly.
The main presidential candidates:
John Magufuli - Chama Cha Mapinduzi
Aged 55, currently works minister
Promised change and to improve on the pace of progress laid down by the previous CCM government
Promised to end power shortages and exploit Tanzania's natural gas discoveries
"My government will put emphasis on fighting corruption, job creation and industrialisation," he said on Saturday
Nicknamed The Bulldozer for driving a programme to build roads across the country.
Edward Lowassa - Ukawa
Aged 62, left CCM when it did not pick him as its presidential candidate earlier this year
Four opposition parties rallied behind him as their joint candidate
"We must stop being a nation of beggars," he told a rally on Saturday. "It is a shame for Tanzania to still be poor after 54 years of independence."
Has served as prime minister, but resigned over corruption scandal in energy sector
Denies involvement in the scandal.
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