Kurtág was born at Lugos (Lugoj in Romania) on 19 February 1926. From 1940 he took piano lessons from Magda Kardos and studied composition with Max Eisikovits in Timisoara. Moving to Budapest, he enrolled at the Academy of Music in 1946 where his teachers included Sándor Veress and Ferenc Farkas (composition), Pál Kadosa (piano) and Leó Weiner (chamber music).
In 1957-58 Kurtág studied in Paris with Marianne Stein and attended the courses of Messiaen and Milhaud. As a result, he rethought his ideas on composition and marked the first work he wrote after his return to Budapest, a string quartet, as his opus 1.
In 1958-63 Kurtág worked as a répétiteur with the Béla Bartók Music Secondary School in Budapest. In 1960-80 he was répétiteur with soloists of the National Philhamonia. From 1967 he was assistant to Pál Kadosa at the Academy of Music, and the following year he was appointed professor of chamber music. He held this post until his retirement in 1986 and subsequently continued to teach at the Academy until 1993.
With increased freedom of movement in the 1990s he has worked increasingly outside Hungary, as composer in residence with the Berlin Philharmonic (1993-1994), with the Vienna Konzerthaus (1995), in the Netherlands (1996-98), in Berlin again (1998-99), and a Paris residency at the invitation of the Ensemble Intercontemporain, Cité de la Musique and the Festival d’Automne.
Kurtág won the prestigious 2006 Grawemeyer Award for Music Composition for his '...concertante...'.