Austria's presidential election remains on a knife-edge with all votes from polling stations now counted.
The interior ministry says that Norbert Hofer of the far-right Freedom Party is currently slightly ahead of his rival, Alexander Van der Bellen.
The result will only be decided when hundred of thousands of postal ballots are counted on Monday.
A key campaign issue was Europe's migrant crisis, which has seen asylum-seeker numbers soar.
About 90,000 people claimed asylum in Austria last year, equivalent to about 1% of the Austrian population, and the Freedom Party ran an anti-immigration campaign.
For the first time since World War Two, both the main centrist parties were knocked out in the first round.
The presidency is a largely ceremonial post, but a victory for Mr Hofer could be the springboard for Freedom Party success in the next parliamentary elections, scheduled for 2018.
The presidents of the European Commission and the European Parliament, Jean-Claude Juncker and Martin Schulz, have both expressed concern over a Hofer victory.
Analysis: Bethany Bell, BBC News, Vienna
Austria is split. The soft-spoken, charismatic Mr Hofer, sometimes described as a wolf in sheep's clothing, caused turmoil in Austrian politics when he won a clear victory in the first round of voting in April.
But now his rival, Mr Van der Bellen from the Greens, has caught up. The far right has profited from deep frustration with the established parties of the centre left and the centre right in Austria. And in recent months, it has been boosted further by fears about the migrant crisis.
If Mr Hofer wins, it could have an impact far beyond Austria's borders - possibly giving momentum to far-right and Eurosceptic parties in other EU countries.
According to the interior ministry's final count of votes cast at polling-stations (in German), Mr Hofer took 51.9% to 48.1% for Mr Van der Bellen.
Postal voting accounts for 750,000 ballots, roughly 12% of Austria's 6.4 million eligible voters, said Interior Minister Wolfgang Sobotka.
"None of us wished for this," Mr Hofer said when he and Mr Van der Bellen were interviewed by ORF after the vote on Sunday.
"After all, both of us wanted to have a good night's sleep but it is so exciting. I've been in politics for a long time but I've never experienced an election night like this one."
Whoever won, he said, would have "the job of uniting Austria".
Mr Van der Bellen said that if he were elected president he would be welcome in all member states of the EU.
"I have been pro-European during the five months of campaigning," he said. "I made clear how important the European Union is for freedom, security and prosperity - also in Austria."
In the first round, Mr Hofer secured 35% of the votes, while Mr Van der Bellen polled 21%.
The two rivals had engaged in an angry TV debate earlier in the week, described as "political mud-wrestling" by commentators.
Vying to lead Austria
- Age: 45
- Background: Aeronautical engineer
- Politics: Far-right Freedom Party
- Campaign soundbite: "To those in Austria who go to war for the Islamic State or rape women - I say to those people: 'This is not your home'"
Alexander Van der Bellen
- Age: 72
- Background: Economics professor
- Politics: Former Green Party leader
- Campaign soundbite: "I've experienced how Austria rose from the ruins of World War Two, caused by the madness of nationalism."
Such was the political shock at the far right's first-round win that the Chancellor (prime minister), Werner Faymann, resigned after losing the support of his Social Democratic party colleagues.
The Social Democrats and the People's Party have governed Austria for decades, either alone or in coalition.
At the last general election in 2013, they together won just enough votes to govern in a "grand coalition".
Incumbent President Heinz Fischer, 77, could not run again after two terms in office.