Bebek was born in Sarajevo, SR Bosnia-Herzegovina, Yugoslavia to parents Zvonimir and Katarina. He showed an early interest in music, entertaining his mother's guests by singing songs he heard on the radio. He also experimented with harmonica, but abandoned it in third grade of primary school as he wanted to play guitar and sing along. His teacher, however, discouraged such intentions so Željko ended up playing mandolin instead. He soon became the school's best mandolin player and was allowed to play guitar as a reward.
Bebek spent the next couple of years singing and playing rhythm guitar with the band, helping them become quite prominent locally. As Kodeksi had perisistent problems filling the bass player spot, Bebek recommended young 18-year-old Goran Bregović after seeing him play with Bestije in 1969.
After a falling out with bandmates during the stay in Italy Bebek left Kodeksi in fall 1970 and returned to Sarajevo.
The new year, however, brought more creative stagnation as their repertoire consisted entirely of foreign covers. In December 1971, Bebek received a call-up to report for the mandatory Yugoslav Army service and Novi Kodeksi played their last ever show in Sarajevo's Dom mladih. Twenty six years of age at this point, Bebek also got married and made a decision to leave the music business altogether.
In March 1973, after getting back from the army, Bebek joined Jutro in earnest, but, not being sure about the band's creative and commercial potential, he also got a clerk job at social security administration. Jutro, however, soon started becoming successful and he quit the government job to again devote to music full-time.
Jutro soon transformed into Bijelo dugme.
Almost 30 years old at this point, he became instant country-wide celebrity.
He ended up spending a full decade with the band before eventually leaving in 1984 to fully pursue a solo career.
In 1984, just before officially leaving Bijelo dugme he recorded his second solo album Mene tjera neki vrag.
Bebek had a few major hits throughout his 11 album run. Most of his hits had strong folk influence, including "Oprosti Mi Śto Te Volim", "Da Je Sreće Bilo", "Jabuka I Vino", "Sinoć Sam Pola Kafane Popio" (with lyrics by Bora Đorđević), "Puca Mi U Glavu", "Ćaśu Otrova", Gdje Će Ti Duśa", and "Da Zna Zora".
In 2005 he took part in 3 large farewell concerts of Bijelo Dugme.