Kurz's People's Party (OeVP) came out ahead with 37.5 percent in September 29 elections, but the far-right Freedom Party (FPOe) with who he governed from 2017 until the "Ibiza-gate" scandal in May tumbled 10 points down to 16.2 percent.
A big winner of the elections were the Greens with 13.9 percent, up by 10 points, as climate change became a top voter concern.
This places Kurz, who has driven a hard anti-immigration line, in a tough spot to find a suitable coalition partner.
After President Alexander Van der Bellen tasked him to form the government, Kurz said he would start official talks with all parties this week, insisting fighting illegal immigration continued to be a top priority.
"It will be important to continue the resolute path to fight illegal immigration in Austria and also Europe," he said in a statement broadcast live on national television, standing next to Van der Bellen.
His other priorities are fighting climate change, fending off any economic downturn, offering tax relief and reforming the social welfare system to ensure it is sustainable, even if that's unpopular, he said.
Coalition negotiations are expected to take months.
FPOe leader Norbert Hofer said last week that he didn't see the election result as a mandate to enter a government, but that the situation could be "reassessed" in the "unforeseen situation" that Kurz failed to find a
coalition partner until next year.
One option, which looks increasingly likely, would be for Kurz to govern with the Greens, but he would need to rebrand himself and may anger some of his more right-leaning voters.
Greens leader Werner Kogler also has said he wants to first assess if it makes sense to enter negotiations, saying he would need a "sign of reversal" from Kurz's past policies.
The OeVP's partner for decades in the past, the Social Democrats (SPOe), fell to a historic low of 21.2 percent, according to official final results, making them also potential partners for Kurz but leaving them much weakened.
The liberal NEOs took 8.1 percent of the vote.
Kurz became the world's youngest elected leader in 2017, but his government fell apart in May after his vice-chancellor, FPOe leader Heinz-Christian Strache, was seen in hidden video footage appearing to offer public contracts in exchange for campaign help to a fake Russian backer.
Strache, who led the party for 14 years, resigned from all posts amid the so-called "Ibiza-gate" scandal, and the fresh elections were triggered.
Following further, more recent allegations against Strache -- that he abused his party expense account -- that cost the party votes, he announced last week that he is withdrawing from politics and public life.