Osipov was born in Tobolsk (in present-day Tyumen Oblast, Russia). In 1959 he graduated from the Department of Mechanics and Mathematics of the Ural State University (Yekaterinburg, Russia). His teacher was Nikolai Krasovsky, famous scientist and founder of the Ural scientific school in mathematical theory of control and the theory of differential games. From 1961 to 1969 he worked at the Ural State University. From 1970 to 1993 he worked at the Institute of Mathematics and Mechanics of the Ural Branch of the USSR Academy of Sciences (later, of the Russian Academy of Sciences) in Yekaterinburg (from 1986 to 1993 he was the chief of the Institute). In 1971 he defended his second thesis and received the rank of professor in 1973. In 1984 he was elected corresponding member and in 1987, full member of the USSR Academy of Sciences (Division of Machine Engineering, Mechanics and Control Processes Problems).
He has been the President of the USSR Academy of Sciences and its successor, the Russian Academy of Sciences, since 1991, having been reelected in 1996, 2001 and 2006. He is also involved in the Russian government, specifically as a member of the Security Council of the Russian Federation.