Bozay studied at the Békéstarhos College of Music and at Budapest Academy of Music. He taught harmony at Szeged Conservatory and worked for the Hungarian radio, and in 1967 he went to Paris for six months under a UNESCO scholarship. Upon his return to Hungary he turned his concentration to composition. His String Quartet No. 1 brought him international attention when it was performed at the International Rostrum of Composers in 1967. His early works are serial and melodic, and after his stay in Paris he turned to tone clusters and composing with timbre. Bozay's work contains some elements of Hungarian peasant music, including folk rhythms and strophic folk songs.