He began to study music under the guidance of his father - conductor and teacher Alexander Ignatievich Klimov [ru].
[1] In 1951 he entered the Kyiv Conservatory (now the Petro Tchaikovsky National Music Academy of Ukraine), where he studied under B.S.
He performed with the largest orchestras in the world under the direction of conductors including Yevgeny Svetlanov, Kirill Kondrashin, Gennady Rozhdestvensky, Konstantin Ivanov, Arvīds Jansons, Leonard Bernstein (USA), Eugene Ormandy (USA), Carlo Zecchi (Italy), Wolfgang Sawallisch (Germany), Norman Del Mar (Great Britain), Serge Baudo (France), Franz-Paul Decker (Canada), and others.
Klimov had numerous records on phonograph and compact disc made by Melodia (Russia), EMI Elektrola (Great Britain), Ariola (Germany), Toshiba (Japan), Victor Company and Angel Records ( USA), "Le Chant du Monde" (France), etc.
His students included Aiman Musakhodzhaeva, Gernot Winischhofer [de], Elena Denisova, Evgeny Bushkov, and many others.