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".
Abbado was 80 years old and had been gravely ill for several months after surviving a stomach tumour in 2000, forcing him to cancel recent performances.
His family said Abbado died "peacefully" at 8:30 am (0730) surrounded by his loved ones in Bologna, where he was the artistic director of the Orchestra Mozart.
There will be a lying-in-state in a church in the city starting on Tuesday and the family has asked mourners to make donations to a local children's cancer ward or a prison instead of bringing flowers and wreaths.
"This is such a painful moment," said a tearful Attilia Giuliani, head of the "Abbadiani" appreciation club in Milan and one of the first to receive the news in a message from his doctor.
The Berlin Philharmonic mourned "an immense and heavy loss" saying his "love of music and his insatiable curiosity have been a source of inspiration for us".
The orchestra's director, Simon Rattle, said that in Abbado's sickness-stricken final years "all the facets of his art came together in an unforgettable way".
The Vienna State Opera's chief Dominique Meyer said the music world "has lost one of its very greats".
The Lucerne Festival in a statement underlined Abbado's "distinctive understanding of the chamber music-like aspect of music-making, his love for the ensemble".
"He wasn't a man of big words, nor was he a fan of long rehearsal discussions. Instead, the artistic process that took shape when he conducted consisted of a kind of silent close listening," it said.
The festival said Abbado's last performance before his death was of Anton Bruckner's Ninth Symphony in Lucerne on August 26, 2013, and was an "unforgettable evening".
Abbado was "deeply transfigured" that night, it said.
Fellow Italian maestro Riccardo Muti, director of the Chicago Symphony Orchestra, said Abbado had "marked the history of conducting for several decades".
Israeli-Argentine conductor Daniel Barenboim said he had been "one of the few to have a close relationship with the spirit of music".
Born on June 26, 1933, in Milan into a musical family, Abbado began his studies in his hometown and completed his training with Hans Swarowsky in Vienna from 1957 -- the start of a long love story with Austria.
'I am not their boss'
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.
He was known as a leftist and broke with tradition by giving concerts in factories and schools, trying to involve the wider public in the classical music world.
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.
He was at the Staatsoper in Vienna between 1986 and 1991 and was elected head of the Berlin Philharmonic by its members in 1989 after the death of Herbert von Karajan, establishing close ties with the orchestra.
"I am not their boss, we work together," he used to say.
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