Davis was born in Ohio in 1944. He later moved to New York to study piano and conducting at the Juilliard School of Music; in 1968 he began to conduct the school's ensemble. He became the musical director for the Saint Paul Chamber Orchestra in 1972.

In 1980, Davies moved to Germany, but continued to perform in the United States, as well as other countries. He has served as conductor or music director for the Stuttgart Chamber Orchestra, the Basel Symphony Orchestra, the Brooklyn Philharmonic Orchestra, the Vienna Radio Symphony Orchestra, the Beethovenhalle Orchestra, the Cologne Philharmonic, the Berlin Philharmonic, the Orchestre de la Suisse Romande, the Munich Philharmonic, and others.

In 1982, Davies saw the release of an album, Ritual, recorded under the ECM Records label. That was far from his first recording though, since he has appeared on albums as conductor for many artists, including Philip Glass, John Zorn, Duke Ellington, and Dave Brubeck.

The prominent ECM recording Abii ne Viderem, won Davies the Japan Record Academy Award for Modern Music in 1995.

Davies had his successful Metropolitan Opera debut in 1996 with a splendid production of Philip Glass' The Voyage. Davies has also conducted Virgil Thompson's Four Saints in Three Acts, Wagner's Flying Dutchman, Stravinsky's The Rake's Progress, William Bolcom's A View From the Bridge, Beethoven's Fidelio, and Verdi's Otello. During his remarkable career, Davies has appeared at the Bonn Opera, the Lyric Opera of Chicago, Avery Fisher Hall, Boston Symphony Hall, Lincoln Center, and Carnegie Hall. He been a strong advocate for the music of Philip Glass and conducted the premiere of the composer's Symphony No. 9 in 2012.