Born on 14 April 1989 in Užice, Serbia, he inherited the football “bug” from his father, an ardent fan of FK Partizan. He grew up in a… black-and-white house and wanted to go into football ever since he was a little kid.

He made his first football steps at local FK Sloboda Užice. He was spotted by FK Partizan’s network in 2005 and Živko’s dream started becoming reality, as he joined the academy of the Belgrade club.

He earned his promotion to the senior team in 2007 and penned his first professional contract. But the road towards establishing himself was all but a smooth one.

He was shipped on loan to FK Teleoptik. He returned to FK Partizan, but then was loaned to FK Metalac Gornji Milanovac, helping them to get promoted to the top tier in 2008-09 and remain in the category in the following season.

Upon his return to Belgrade, he was FK Partizan’s third goalkeeper. After a season spent without any action, it was decided that he would leave on loan to FK Metalac Gornji Milanovac once again.

He started the 2012-13 season at FK Teleoptik, but Stojnović’s injury led FK Partizan to terminate Živković’s loan contract and bring him back to Belgrade.

The Serbian goalkeeper worked hard for a place in the starting line-up. Despite hardship, he didn’t give up and got vindicated in spring 2015. Živko became the starting keeper, lined up a series of excellent performances and kept his place in the line-up for almost a year. He then fell victim of the coaching change.

In the end of 2015-16 season, his contract with FK Partizan expired and he decided to continue his career elsewhere. Xanthi’s offer seemed ideal for him –and proved so.

Živković spent three years with the Thrace outfit, played in 89 encounters and has been established as one of the Greek Super League’s best goalkeepers.

In April 2019 he announced via social media that he wouldn’t remain at Xanthi. PAOK didn’t let the opportunity escape and signed him on a free transfer.