Zelmanov was born into a Jewish family in Khabarovsk, Soviet Union (now in Russia). He entered Novosibirsk State University in 1972, when he was 17 years old. He obtained doctoral degree at Novosibirsk State University in 1980, and a higher degree at Leningrad State University in 1985. He had a position in Novosibirsk until 1987, when he left the Soviet Union.
In 1990 he moved to the United States, becoming a professor at the University of Wisconsin–Madison. He was at the University of Chicago in 1994/5, then at Yale University. As of 2011, he is a professor at the University of California, San Diego and a Distinguished Professor at the Korea Institute for Advanced Study.
Zelmanov was elected as a member of the U.S. National Academy of Sciences in 2001, becoming, at the age of 47, the youngest member of the mathematics section of the academy. He is also an elected member of the American Academy of Arts and Sciences (1996) and a foreign member of the Korean Academy of Science and Engineering and of the Spanish Royal Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society.
Zelmanov gave invited talks at the International Congress of Mathematicians in Warsaw (1983), Kyoto (1990) and Zurich (1994). He was awarded Honorary Doctor degrees from the University of Alberta, Canada (2011), Shevchenko National University of Kyiv, Ukraine (2012), the Universidad Internacional Menéndez Pelayo in Santander, Spain (2015) and the University of Lincoln, UK (2016).
Zelmanov's early work was on Jordan algebras in the case of infinite dimensions. He was able to show that Glennie's identity in a certain sense generates all identities that hold. He then showed that the Engel identity for Lie algebras implies nilpotence, in the case of infinite dimensions.