Abbado believed deeply in what he called the "therapeutic values" of music and staged performances on factory floors of the 1960s, as well as promoting the Simon Bolivar Symphony Orchestra from Venezuela.
After being nominated a senator for life in the Italian parliament last year, Abbado gave away his senator's salary to provide scholarships for young musicians saying that music "helps people live better together".
"Claudio Abbado died peacefully this morning at 8:30 am local time (1300 IST) surrounded by his family" in Bologna in central Italy where he had been artistic director of the Orchestra Mozart, his relatives said in a statement.
Abbado made hundreds of recordings with Deutsche Grammophon, Sony and EMI, spanning from Italy's lyrical repertoire to contemporary classical music.
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"This is such a painful moment," said a tearful Attilia Giuliani, head of the "Abbadiani" appreciation club in Milan, who said she first saw the conductor perform at the San Carlo opera house in Naples in 1988.
He once said that he felt he was "half Viennese".
Abbado was a true Europhile and a global citizen who worked in many of the world's musical capitals.
The maestro started out at La Scala in 1960, where he was widely praised for his performance of Giacomo Manzoni's opera "Atomtod" in 1965 and he served as musical director of the celebrated theatre until 1986.
After a trip to Venezuela, he became a major supporter of maestro Antonio Abreu's award-winning Simon Bolivar Symphony Orchestra for disadvantaged children.
In Venezuela "there are hundreds of orchestras for young people and the music really rescues them from criminality, prostitution and drugs," Abbado said.
From 1971, he also became a regular at the Vienna Philharmonic and he conducted the London Symphony Orchestra between 1979 and 1988, where he was praised for concerts of his favourite composer, Gustav Mahler.