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Leviev graduated from the Bulgarian State Academy of Music in 1960 majoring in composition under Pancho Vladigerov and in piano under Andrei Stoyanov. As a student, he won second prize at the International Competition in Vienna for his Toccatina for piano. His professional development as a composer began at the Drama Theatre in Plovdiv. He was appointed conductor of the big band of Bulgarian National Radio after Emil Georgiev left office (1962–66). His ideas were innovative in the orchestra; pieces like Studia, Blues in 9 or Anti-waltz became a standard of a successful synthesis of folklore and jazz. From 1963 to 1968 he worked as soloist and conductor of the Sofia and Plovdiv Philharmonic Orchestra. In 1965, embracing the idea of the writer Radoy Ralin, he founded Jazz Focus '65 with which he toured until 1970, achieving success for Bulgarian jazz and winning prizes at jazz festivals in Montreux, Sofia, and Prague.
In 1970 he left Bulgaria for political reasons and moved to Los Angeles. He was allowed to perform in Bulgaria not earlier than 1980.
He worked as composer, arranger, and pianist for Don Ellis (1970–1975) Orchestra and the Billy Cobham Band (1971–77). He toured the U.S. and Europe; he was Music Director of Lainie Kazan (1977–80). He gave concerts and made recordings with John Klemmer, Art Pepper, and Roy Haynes. He toured Europe with Pepper (1980–82) and was one of the founders of the fusion band Free Flight. In 1983 he became music director of the Jazz Sessions at the Comeback Inn in Venice, California. He gave concerts in Japan with bassist Dave Holland (1983–86) and organized solo jazz recitals in Europe (1985–86). He has taught jazz composition at the University of South California and given master classes at the New Bulgarian University in Sofia.
He has composed symphony and chamber works, big band, and jazz orchestra music. In the 1960s he wrote film music.
In 1987 he won the Drama-Logue Award for best music director given by the Pacific Conservatory of the Performing Arts. His works were recorded by Atlas, Alpha, Discovery, Balkanton, Columbia, Atlantic, and ABC. Part of his works were published by Dick Grove Publications and the Bulgarian publishing house Nauka I Izkustvo (up to 1968). He won a prize at the National Jazz Educational Congress, a Grammy Award for arrangement, and the Honorary Gold Medal of the Académie internationale des Arts in Paris (1995). He was awarded an honorary Doctor Honoris Causa by the Academy of Music, Dance and Fine Arts in Plovdiv (1995) and by the New Bulgarian University.
- Jazz Focus (Saba, 1968)
- Music for Big Band and Symphony Orchestra (Trend, 1977)
- Blue Levis (Dobre, 1980)
- What's New (Atlas, 1980)
- Plays the Music of Irving Berlin (Discovery, 1982)
- True Blue (Mole, 1982)
- Up and Down (MA, 1987)
- Easter Parade (Discovery, 1988)
- Bulgarian Piano Blues (MA, 1989)
- Live at Vartan Jazz (Vartan Jazz, 1994)
- Atmospheric Conditions Permitting (Decca/ECM, 1995)
- Leviev-Slon & Co. Play the Beatles: When I'm 64 (City Hall, 1999)
- The Man from Plovdiv (MA, 1999)
- A Voyage Again/Con Mucho Gusto (CDBaby/N' House, 2006)
- Multiple Personalities: Milcho Leviev Plays the Music of Don Ellis (Mighty Quinn, 2006)
With Billy Cobham
With Don Ellis
With Roy Haynes
With Al Jarreau
- Breakin' Away (Warner Bros., 1981)
With Arif Mardin
- Journey (Atlantic, 1974)
With Airto Moreira
- Virgin Land (Salvation, 1974)
With Art Pepper
- Blues for the Fisherman (Mole, 1980)
With Gerald Wilson