Myanmar's opposition National League for Democracy says it is confident of victory in the first openly contested national election in 25 years.
An NLD spokesman said it expected to win about 70% of seats. Party leader Aung San Suu Kyi said: "I think you all have the idea of the results."
Official results have been released for only 54 seats, with 48 won by the NLD.
The military-backed Union Solidarity Development Party (USDP) has been in power since 2011.
The NLD says it has won 44 out of Yangon's 45 seats in the lower house, and 70% of seats nationally, but this has not been confirmed.
More than 6,000 candidates from more than 90 parties were vying for 498 seats in both houses of parliament.
NLD spokesman Win Htein said it had "accumulated proof that there are some deeds by the authorities which are against the election regulations".
The acting chairman of the USDP has told BBC Burmese that he has lost his own seat in the constituency of Hinthada to the NLD - seen as a key indicator of election results.
"We have to find out the reason why we lost," U Htay Oo said. "However, we do accept the results without any reservations. We still don't know the final results for sure."
Earlier, Ms Suu Kyi addressed a crowd at the NLD's headquarters in Yangon, urging them to be patient.
A quarter of the parliamentary seats are reserved for the army, and for the NLD to have the winning majority it will need at least two-thirds of the contested seats.
The BBC's Jonah Fisher in Yangon says that while this is a very big ask, it is by no means impossible if the party, which is popular in urban areas, manages to win seats in rural areas which tend to be dominated by ethnic minorities.
But Ms Suu Kyi cannot become president because the constitution bars anyone with foreign children from holding the post. Her two sons, with her late husband, are British.
Our correspondent says that if the NLD win, it will face difficulties in changing the constitution on its own as the document still gives the military considerable power, and the party would most likely nominate someone else to be president. Ms Suu Kyi has said she would be "above the president".
Read more: Four scenarios for Myanmar's crucial vote
At the scene: the BBC's Jonah Fisher in Yangon
For now no one is crying foul over the slow release of results.
Aung San Suu Kyi and her party are being patient, confident that they're on the brink of a historic win.
Results have to be verified at both a regional and national level so there's a fair amount of bureaucracy involved.
That has not stopped them declaring their own results. To the delight of the crowd the NLD announced sweeping victories across Yangon, Mon State and the Irrawaddy Delta.
The percentages involved would make even an African despot blush. It mattered little to the crowd that they weren't official figures.
Over the next few days the steady flow of real results is expected to continue.
That slow pace may give people time to come to terms with what is looking like being a massive political change.
Myanmar's historic election
In pictures: Election day
Decision-making in the Delta: Jonathan Head on the small but crucial town of Hinthada
Aung San Suu Kyi, profiled: International symbol of peaceful resistance
'Abandoned people': What rights do the Rohingya Muslims have?
Elections explained: Why does this vote matter?
Tens of thousands of officials and volunteers have been counting the votes, first in each of the 50,000 polling stations, then tallying them in constituency offices of the Election Commission.
In one of the earliest and most significant known results, the ruling party's Shwe Mann, who is Speaker of the lower house of parliament, conceded defeat in his constituency to the NLD candidate.
The full results will not be known for at least a few days, and the president will only be chosen in February or possibly later.
International observers say the voting process was generally smooth, with some isolated irregularities.
And hundreds of thousands of people - including minority Rohingya Muslims - were also denied voting rights, raising concerns about the fairness of the poll.
US Secretary of State John Kerry hailed the elections as a step towards democracy, but added that they were far from perfect.
About 30 million people were eligible to vote in Sunday's election in Myanmar, which is also known as Burma.
Turnout has been estimated at about 80%, in what were the first national elections since a nominally civilian government took power in 2011.
"I'm really happy because from what I heard the NLD is winning. I couldn't sleep until 11 or 12 because I was looking everywhere for results," San Win, a 40-year-old newspaper vendor, told the AP news agency.
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