Born in Szőreg, a southeastern suburb of Szeged in southern Hungary, Bugner and his family fled after the 1956 Soviet invasion and settled in Britain. Standing at with a prime weight of around , Bugner twice held the British and British Commonwealth heavyweight titles and was a three-time European heavyweight champion. He was ranked among the world's top ten heavyweights in the 1970s, fighting such opponents as Muhammad Ali, Joe Frazier, Ron Lyle, Jimmy Ellis, Manuel Ramos, Chuck Wepner, Earnie Shavers, Henry Cooper, Brian London, Mac Foster, Rudie Lubbers, Eduardo Corletti, Jurgen Blin and George Johnson. He fought for the world heavyweight championship in 1975, losing on points in a second bout with Ali.