He was co-founder of the legendary Eisenstein Archives, and their director from 1967 to 1985. In 1989 he founded the Moscow Cinema Museum, the Musei Kino, whose director he became in 1992. Since 1988 he teaches at the New York University, as well as at the German Film and Television Academy Berlin and gives lectures worldwide.

He has published extensively on film theory and the history of Soviet and Russian cinema. As an expert on Sergei Eisenstein he wrote on his work and helped with the reconstruction of his film Bezhin Meadow, The General Line und The Battleship Potemkin.

Kleiman has received numerous awards, including a FIPRESCI Prize for the retrospectives he organised during the Moscow International Film Festival, a "Felix" from the European Film Academy, as well as a Goethe Medal. He has been on juries at many film festivals, such as Venice, Leipzig, Toronto, Locarno, including the International Jury of the Berlinale Competition. For years he was also an advisor and active supporter of the Berlinale's Forum section.