His father was the renowned violinist and music teacher Michelangelo Abbado. He is the brother of the composer Marcello Abbado and therefore uncle of the conductor Roberto Abbado. After school, Abbado studied at the Giuseppe Verdi Conservatory in Milan. From 1956 to 1968 he was married to Giovanna Cavazzoni. Together they became parents to the theater manager Alessandra Abbado and the opera director Daniele Abbado. In 1957 he moved to Vienna, where he went to school with conductor teacher Hans Swarowsky. In 1968, Abbado began working for the Milan opera house "Scala", where he worked in various functions until 1986. The conductor was committed to opening up the so-called "high culture" to the common people, who had once been the audience and protagonists of traditional Italian opera.

From 1979 to 1988, Abbado was also principal conductor of the London Symphony Orchestra. Replaced by Riccardo Muti at the helm of La Scala in Milan, Abbado moved to Vienna in 1986 to take over the management of the State Opera until 1991. From 1990 to 2002 he was chief conductor of the Berlin Philharmonic. However, cancer interrupted Abbado's artistic career in the late 1990s. After his recovery, the internationally celebrated conductor only appeared at concerts sporadically. In spring 2004, Abbado's guest appearance with the Berlin Philharmonic was a great success. In the spring of 2005, Abbado conducted Mozart's "Magic Flute" for the first time, with which he celebrated great success in Baden-Baden and Italy. At the end of May 2005, the maestro gave three concerts in Berlin to sold-out houses.

Abbado was considered a supporter of young musicians, for whom he founded the European Union Youth Orchestra in 1978, the Gustav Mahler Youth Orchestra in 1988, the Mahler Chamber Orchestra in 1997 and finally the Mozart Orchestra in 2004. The youth orchestras support the training of young talent with scholarships. In 2003 he was honored with the Praemium Imperiale. In August 2005, Abbado, after Vienna, was also made an honorary citizen of Lucerne, whose music festival he was closely associated with. For the 250th year of Mozart's birth, Abbado designed numerous Mozart Orchestra initiatives in 2006. In 2008 he received the Wolf Prize. On August 30, 2013, Claudio Abbado was appointed senator for life by President Giorgio Napolitano. In the same year, his book "My World of Music" was named science book of the year in Austria.

Claudio Abbado died on January 20, 2014, at the age of eighty, in Bologna after a long battle with cancer.