His family traced its roots to a prominent Moorish family expelled from Spain in 1492 and is said to include the architect of the Alhambra. His father was Michelangelo Abbado, a violinist and teacher who gave both Claudio and his brother, Marcello Abbado, their first piano and music lessons (Marcello has gone on to become a pianist and composer).
Claudio was educated at the Milan Conservatory, graduating in 1955 with a certificate in piano. While a student there, he also studied conducting with Antonio Votto. In 1955, he studied piano with Friedrich Gulda in Salzburg, and then (from 1956-1958) attended Hans Swarowsky's conducting classes at the Vienna Academy of Music. In 1958, he made his conducting debut in Trieste, and won the Koussevitzky Competition at Tanglewood. This led to his engagement in several provincial opera houses and orchestras, as well as a teaching position at Parma Conservatory. After winning the 1963 Dimitri Mitropoulos Prize, he was awarded a five-month position on the staff of the New York Philharmonic, with which he made his American professional debut on April 7, 1963. The year 1965 marked his debut with the Vienna Philharmonic at the Salzburg Festival, but more so for his return to his hometown of Milan to conduct at La Scala. He would conduct there again in 1967, become the company's resident director in 1968, and be named music director in 1971. Abbado was credited with raising the performing standards at La Scala, and for broadening its repertory. He also organized its orchestral players into the Orchestra della Scala, whose performances of works by such composers as Schoenberg, Nono, Ligeti, Stockhausen, and Berio brought Abbado special acclaim as an interpreter of modern works. In 1971, he was appointed principal conductor of the Vienna Philharmonic. Soon after, he formed a relationship with the London Symphony Orchestra, of which he became principal conductor in 1979, and finally music director, a post he retained until 1988. Other activities during this period included the founding of the European Community Youth Orchestra, the Chamber Orchestra of Europe, and the Gustav Mahler Youth Chamber Orchestra. For the period 1982-1986, Abbado served as principal guest conductor at the Chicago Symphony Orchestra, after which he was appointed music director of the Vienna State Opera. The Viennese further honored him by appointing him general music director of the city. During his tenure, he founded (1988) the Wien Modern music festival, which has now grown to encompass all forms of contemporary art. Abbado succeeded Herbert von Karajan as music director of the Berlin Philharmonic in 1989. In 1991, he relinquished his position at the Vienna State Opera, but he has maintained a presence in the city by founding the annual Vienna prize for young composers. In 1994, he succeeded to another post once held by Karajan, namely the directorship of the Salzburg Easter Festival. Following a bout with cancer in 2000, Abbado formed the Lucerne Festival Orchestra in 2003 with the orchestra's artistic and executive director, Michael Haefliger. Though Abbado briefly suspended his activities in 2007, he took up the baton again after a two month absence. In later years, Abbado reduced his workload due to illness, but still found time to conduct concerts and record with his hand-picked Orchestra Mozart, and worked with several youth orchestras across Europe.