Korkut has 42 caps and 1 goal for the Turkish national team. He represented his country at the UEFA Euro 1996 and UEFA Euro 2000.

Korkut worked as a coach for Real Sociedad U19, 1899 Hoffenheim U17, and VfB Stuttgart U19 before serving as an assistant coach for the Turkish national team. He received the UEFA Pro License from the Cologne Sports Academy.

Hannover 96
He was appointed as the head coach of Hannover 96 on 31 December 2013. He became the second Turkish Bundesliga head coach ever, after Özcan Arkoç. His first match was a 3–1 win against VfL Wolfsburg on 25 January 2014. Hannover finished the 2013–14 season in 10th place. Hannover started the 2014–15 season. He was sacked on 20 April 2015. His final match was a 4–0 loss to Bayer Leverkusen. Hannover was in 15th place at the time of the sacking. He finished with a record of 15 wins, 11 draws, and 22 losses.

FC Kaiserslautern
On 17 June 2016, he was selected as the head coach of 1. FC Kaiserslautern. He resigned on 27 December 2016.[14] He finished with a record of four wins, seven draws, and seven losses.

Bayer Leverkusen
On 6 March 2017, he was appointed as the new coach of Bayer Leverkusen until the end of the season. One week before the end of the season, it was decided that Korkut's contract would not be extended. He finished with a record of two wins, six draws, and four losses.

VfB Stuttgart
On 29 January 2018, Korkut was appointed as coach of VfB Stuttgart. In June 2018, he extended his contract with Stuttgart until June 2020. He was sacked on 7 October 2018.

Hertha BSC
On 29 November 2021, Korkut was named head coach of Hertha BSC.