[3] Leviev graduated from the Bulgarian State Academy of Music[4] in 1960, [5] majoring in composition under Pancho Vladigerov and in piano under Andrei Stoyanov.
After Emil Georgiev left office, Leviev was appointed conductor of the Bulgarian National Radio big band (1962–66).
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).
[5] 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.
He won in the Best Modern Instrumental Album category at the Armenian Music Awards in 2002 for scoring the Nigol Bezjian film, Chickpeas.