Slatkin was born into a family of musicians. His father, Felix Slatkin, was the violinist, conductor and founder of the Hollywood String Quartet and his mother, Eleanor Aller, performed in the quartet as cellist.
He studied at Indiana University and Los Angeles City College before joining the Juilliard School where he studied conducting with Jean-Paul Morel. He made his debut as conductor in 1966 and became assistant conductor of the Saint Louis Symphony Orchestra in 1968. He remained there until 1977, when he became principal conductor of the New Orleans Philharmonic Orchestra. He performed for the first time with the New York Philharmonic in 1974 and regularly conducted the largest American orchestras.
Slatkin returned to Saint Louis in 1979 as musical director of the symphony orchestra. The orchestra's national profile greatly improved under his leadership. He remained in this position until 1996, when he became conductor of the National Symphony Orchestra in Washington. His contract with this orchestra was scheduled to end in 2008 and he then became the new music director of the Detroit Symphony Orchestra. Among other things, he succeeded Sir Andrew Davis as chief conductor of the BBC Symphony Orchestra from 2000 to 2005.
In 2011, he became conductor of Lyon National Orchestra.