Barred from the top spot

And yet, the Nobel Peace laureate -- the leader of Myanmar's long-fought democracy movement -- can't become president.

A change in Myanmar's constitution, drafted by the military, prevents anyone with foreign family members from becoming the nation's leader.

Suu Kyi's late husband was British. Her children hold British passports.

Still, she's pushing forward.

In letters published in local media, Suu Kyi had requested a joint meeting with the commander in chief of Myanmar's armed forces, the chairman of the parliament and Thein Sein.

She said last week she would be "above the President" if her party won the parliamentary election.

But complicating any efforts to change the rules in the future, the military also has an effective veto over any proposed constitutional changes.

READ: Anti-Muslim tension simmers in Myanmar

Election results

It will take at least two weeks to tabulate all the results, a presidential spokesman told CNN.

Already, the military-aligned ruling party has admitted it has lost more seats than it has won. But the scale of its losses and the NLD's victories is still emerging.

"As far as we know from the early signal, there is a majority for NLD in the coming parliament," said presidential spokesman Ye Htut.

"I, on the behalf of President U Thein Sein, want to congratulate Aung San Suu Kyi and the NLD for their success in the election and wishing they can fulfill the desire of the Myanmar people for the big change in the future."

Military still holds many of the cards

The landmark election is seen as a test of the powerful military's willingness to let the country continue along a path toward full democracy, after decades of military-dominated rule in Myanmar, also known as Burma.

The ruling party of Thein Sein has promised the outcome of Sunday's vote will be respected, but the system is already configured strongly in favor of the military, which gets to appoint a quarter of all lawmakers in the two houses of parliament.

What's at stake in Myanmar's elections?

The public is electing 168 of the 224 representatives in the upper house of the national parliament, with the remaining quarter of the seats reserved for lawmakers appointed by the military.

In the lower house, 325 of the 440 seats are up for grabs. Another 110 are reserved for military appointees, while voting has reportedly been canceled in the remaining five electable lower house seats because of security concerns.

Free and fair?

The changes ushered in under Thein Sein since 2011 have helped reduce the country's international isolation, with Western sanctions being eased and foreign investment starting to ramp up.

But human rights groups have warned more recently of a rise in politically motivated arrests, as well as discrimination directed against the Muslim minority, notably the stateless Rohingya population.

Questions have come up over how free and fair Sunday's election will turn out to be. Suu Kyi, the daughter of an independence leader, expressed concern last week about irregularities in advance voting, fraud and intimidation.

Many people still remember the last national election her party contested, in 1990. It was widely considered to have won that one, but the military rulers annulled the results and placed Suu Kyi and many of her colleagues under arrest.

She spent much of the next two decades under house arrest, becoming an internationally recognized symbol of democracy and the country's most popular politician.

Voters today have more awareness

On Tuesday, she told the BBC that she doesn't expect a repeat of 1990.

"The times are different. The people are different," Suu Kyi said, describing citizens as "very much more alert to what is going on around them."

Nonetheless, hundreds of thousands of people in Myanmar are disenfranchised, including Rohingya Muslims in the west of the country, who are denied citizenship, and residents of conflict zones where the election commission canceled voting.

How Myanmar election could affect Hillary Clinton's campaign

After the outcome of the parliamentary vote is decided, lawmakers will begin the complex process of choosing a president.

U.S. Secretary of State John Kerry has described Sunday's elections as "an important step forward" but also cautioned that what happens next is key to Myanmar's future.

"A peaceful post-election period is crucial for stability and maintaining the confidence of the people in the credibility of the electoral process and the overall political transition," he said in a statement.