Davis was born in Ashridge in Hertfordshire, and studied at the Royal College of Music, King's College, Cambridge and then with Franco Ferrara in Rome.
The first major post he held was as associate conductor of the BBC Scottish Symphony Orchestra, a post he took in 1970. In 1975, he took the job of principal conductor with the Toronto Symphony Orchestra. He stepped down in 1988, but has remained close to the orchestra as its Conductor Laureate.
In 1988, Davis became director of Glyndebourne, and the following year became returned to the BBC to become principal conductor of the BBC Symphony Orchestra, a job he held until 2000, when Leonard Slatkin took over.
In 2000, Davis became director of the Chicago Lyric Opera.
Davis has performed quite a wide range of repertoire, though his support of contemporary British music is of particular note. He is particularly associated with Michael Tippett, conducting the British premiere of his work The Mask of Time. He has also made a critically acclaimed recording of Harrison Birtwistle's opera, The Mask of Orpheus.