György Kurtág was born to Jewish Hungarian parents. From the age of 14, he took piano lessons from Magda Kardos and studied composition with Max Eisikovits in Timișoara. He moved to Budapest in 1946 and became a Hungarian citizen in 1948. There, he began his studies at the Franz Liszt Academy of Music, where he met his wife, Márta Kinsker, as well as composer György Ligeti, who became a close friend. His piano teacher at the academy was Pál Kadosa. He studied composition with Sándor Veress and Ferenc Farkas, chamber music with Leó Weiner, and theory with Lajos Bárdos, and graduated in piano and chamber music in 1951 before receiving his degree in composition in 1955.

Following the Hungarian uprising in 1956, Kurtág's time in Paris between 1957 and 1958 was of critical importance for him. There, he studied with Max Deutsch, Olivier Messiaen, and Darius Milhaud. During this time, however, Kurtág was suffering from severe depression. Kurtág received therapy from art psychologist Marianne Stein, who encouraged him to work from the simplest musical elements, an encounter that revivified him and strongly stimulated his artistic development. During this time, he also discovered the works of Anton Webern and the plays of Samuel Beckett. The string quartet he composed in 1959 after his return to Budapest marks this crucial turning point; he refers to this piece as his Opus 1. He dedicated it to Stein.

Kurtág worked as a répétiteur at the Bartók Music School (1958–63) and at the National Philharmonia in Budapest (1960–68). In 1967, he was appointed professor of piano and later also of chamber music at the Franz Liszt Academy, where he taught until 1993. During this time his students included Zoltán Kocsis and András Schiff.

Kurtág's first international opportunity came in 1968 when his largest work to date, The Sayings of Peter Bornemisza, was performed by Erika Sziklay and Lóránt Szűcs at the Darmstadt Summer Courses for New Music. His international recognition began to grow later with Messages of the Late Miss R.V. Troussova for soprano and chamber ensemble, which had its premiere in Paris in 1981.

Since the early 1990s, he has worked abroad with increasing frequency: he was composer in residence at the Berlin Philharmonic (1993–95) and the Vienna Konzerthaus Society (1995). He then lived in the Netherlands (1996–98), again in Berlin (1998–99) and upon invitation by Ensemble InterContemporain, Cité de la Musique, and Festival d'Automne, in Paris (1999–2001). Kurtág and his wife lived near Bordeaux from 2002 to 2015, when they moved back to Budapest.

In 2011, the new director of the Salzburg Festival Alexander Pereira announced that he had commissioned an opera from him, based on a text by Samuel Beckett Endgame, intended to be created during the 2013 edition of the Festival. However, the composition was not completed in time, and the opera was rescheduled for 2015, still in Salzburg. At the end of October 2014, he announced that his wife was seriously ill and he would not be able to complete the work on time. The creation is therefore replaced by a new production of Die Eroberung von Mexiko by Wolfgang Rihm and, at La Scala which was to co-produce the show, Wozzeck by Alban Berg. Fin de Parti was finally created at La Scala in 2018.