Muamer Tanković was born to Bosnian Muslim parents in Norrköping and grew up in Hageby, a Million Programme area in the southern part of the city.

[8] Tanković started his first league match against Manchester United on 9 February 2014 at Old Trafford, and was substituted at half-time for Darren Bent as the game finished 2–2.

[14] On 6 October 2014, in a 2–2 game against Twente, coming on as a sub, he sparked some controversy when celebrating a late equalizer by running out to the sideline and taking a seat at the bench.

[15] In the fourth round of the KNVB Cup, in December 2014, Tanković scored a brace against NEC as AZ won 2–1 and qualified for the quarter-finals.

[18] Tanković appeared in 7 of AZ's games in the 2015–16 Europa League, where the club got knocked out in the group stage after finishing at the foot of the table.

[24] Weeks later, Tanković scored a long range strike against the Russian outfit Zenit Saint Petersburg in a 3–2 win in the 2016–17 Europa League.

[30] During the fall, Tanković struggled to produce points and received heavy criticism from pundit Alexander Axén,[31] before scoring his first league goal for Hammarby in a 1–3 loss against Halmstad on the ultimate match day of 2017.

On 31 July, his contract with Hammarby expired and he reportedly attracted interest from clubs such as Queens Park Rangers, AEK Athens and Beşiktaş.

[40] On 29 October 2020, he scored with a close shot in a 2–1 home defeat against Leicester City for the second game of the Europa League group stage.

[42] On 20 December 2020, he scored with a direct corner kick, in a 2–2 home draw against Volos, becoming the second player to do so, after Levi García a week earlier.

[49] Tankovic made his competitive international debut in a UEFA Euro 2020 qualifying game against the Faroe Islands national football team on 18 November 2019, which Sweden won 3–0.

[50] Tanković is of Bosnian descent, and while playing for Sweden in the Euro Under-19 Championship qualifiers against Bosnia and Herzegovina, he scored a hat-trick but did not celebrate out of respect for his parents' homeland.

[54] Newspaper The Independent described Tanković's ability as "effective", due to his main forward position, his key attributes – guile, creativity and vision, as well as, his "excellent movement that it make him hard to track as he operates between the lines.