It is the first time a vote has taken place under Kenyan new constitution, designed to prevent a repeat of violence that followed the 2007 polls.
More than 1,000 people died in widespread ethnic violence when supporters of rival candidates clashed.
Despite appeals for calm, at least four police officers died when they were attacked near Mombasa on Monday.
At least six other people – including several attackers – are also reported to have died in the assault in the early hours in Changamwe, half an hour’s drive from the centre of Mombasa.
Reports from around the country suggested long lines of voters were forming before polling stations opened.
Some technical difficulties were reported with newly instituted biometric voting kits – designed to counter claims of vote-rigging and long delays in announcing poll results that were partly blamed for the violence last time.
Kenyans will choose a president, members of parliament and senators, county governors and members of 47 county assemblies.
Eight presidential candidates are standing but it is essentially a two-horse race pitting Prime Minister Raila Odinga against Deputy Prime Minister Uhuru Kenyatta.
Some observers say they are particularly concerned about violence erupting should neither of the two frontrunners poll more than 50% – in which case the vote will go to a run-off, probably on April 11.
Uhuru Kenyatta, son of Kenya’s founding father Jomo Kenyatta, is due to stand trial at the International Criminal Court for his alleged role in orchestrating the violence five years ago.
His running mate, William Ruto, has also been indicted by the court. Both men deny any wrongdoing.
The post-election violence of 2007-8 broke out after Raila Odinga claimed he had been cheated of victory by supporters of President Mwai Kibaki.
Supporters of the rival candidates, from different ethnic groups, took up arms against each other.
Raila Odinga later joined a government of national unity under a peace deal.
In the run-up to Monday’s vote, President Mwai Kibaki – who is not seeking re-election – urged Kenyans to vote peacefully and for the losers to accept defeat.
“Cast your vote and keep the peace,” he said in a televised address to the nation on Friday.
“Let us send a clear message to the world that our democracy has come of age. A peaceful vote is a vote for a secure, prosperous and stable Kenya.”
Candidates have also promised to respect the result and urged their supporters to refrain from violence.
Clerics across Kenya also gave sermons dedicated to peace on Sunday.
The police, however, have warned of conspiracies to cause chaos – in Nairobi and elsewhere – and have made it clear that violence will not be tolerated.
Security is being stepped up with some 99,000 police officers being deployed around the country, at polling stations and vote-counting centres.
The polling stations are open from 06:00 to 17:00 local time.
Presidential candidates must secure support from across the country to be declared the winner, so they cannot just rely on support from their ethnic groups, as has been the case in previous elections.
Official results will be announced by March 11 by the electoral commission.
Kenyan election in numbers:
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