Thousands of Burmese are voting in by-elections, which see Nobel Prize winner Aung San Suu Kyi running for political office for the first time.
Her National League for Democracy (NLD) is competing for all 45 seats, in the first vote it has contested since 1990.
Foreign journalists and international observers are being given the widest access they have ever had in Burma.
The European Union has hinted that it could ease some sanctions if the vote goes smoothly.
Analysts say early indications point to a very high turnout.
EU observer Ivo Belet said voting was going peacefully.
"We hope the whole day can be run in a peaceful way and we'll make an evaluation later on the basis of all the polling sessions that we will be seeing," he said.
BBC correspondent Rachel Harvey says voting appears to have been going smoothly at the polling station she is at in Rangoon but that the NLD is alleging some voting irregularities in the capital, Naypyidaw.
She says the party has claimed that the number of people who have been casting ballots for the military-backed party is implausible and that one polling station in Rangoon opened half an hour early.
But our correspondent says an EU election observer has told the BBC that - in her view - any anomalies are probably a case of inexperience rather than a deliberate attempt to manipulate the vote.
A NLD spokesman told AFP news agency he had sent a letter of complaint to the election commission over allegations ballot forms had been tampered with.
Nyan Win said there had been complaints that wax had been put over the check box for the party, which could later be rubbed off to cancel the vote.
"This is happening around the country. The election commission is responsible for what is occurring," he said.
Burma's current government is still dominated by military and ex-military figures from the old regime that ruled the country for decades and was accused of widespread rights abuses.
But since 2010, when a transition of power began, the government has impressed observers with the pace of change.
Most political prisoners have been freed, media restrictions have been relaxed and, crucially, Ms Suu Kyi and the NLD have been persuaded to rejoin the political process.
They have taken no part in Burma's political process since 1990, when the NLD won a landslide victory in a general election but the military refused to accept the result.
Ms Suu Kyi spent much of the following 20 years under house arrest and refused to take part in the 2010 election, which ushered in the current reforms.
The NLD is one of 17 opposition parties taking part in Sunday's election. Only a fraction of seats are up for grabs and the military-backed party will still dominate.
Ms Suu Kyi, 66, is standing for a lower house seat in the Kawhmu Township constituency outside Rangoon.
On Sunday, Ms Suu Kyi visited polling stations in Kawhmu before heading back to Rangoon.
The BBC's Fergal Keane, who is travelling with her, tweeted that she is still feeling the physical strains of the campaign. Last week Ms Suu Kyi suspended her campaign because of ill-health which aides said was triggered by exhaustion.
Earlier, Ms Suu Kyi described this year's election campaign as not ''genuinely free and fair" and warned that reforms were "not irreversible".
But she said she and the NLD did not regret taking part.
"Still we are determined to go forward because this is what our people want," she said.
A small number of representatives from the Association of South-East Asian Nations (Asean), along with the EU and US, have been invited to observe polling.
More than 100 foreign journalists are believed to have received permission to cover the vote.
EU Trade Commissioner Karel De Gucht said political sanctions on Burma were mostly "aimed towards individuals" and could be eased when EU foreign ministers met in Brussels on 23 April.
The lifting of such sanctions could "even happen with immediate effect", he told AFP news agency.
"I am excited by the prospect that finally, hopefully, Myanmar [Burma] citizens will get more freedom," Mr De Gucht added.
"Political freedoms and economic freedoms always go together."
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