Master of Science, University St. Petersburg, 1976. Doctor of Philosophy, Mathematics Institute Academy of Sciences, St. Petersburg, 1979. Doctor of Science., Mathematics Institute Academy of Sciences, St Petersburg, 1985.
His research interests include algebraic geometry, symbolic computation and computational complexity theory in computer algebra, with over 130 published articles During 1976–1992 he was with LOMI, Leningrad Department of the Steklov Mathematical Institute of the Union of the Soviet Socialist Republics Academy of Sciences. In 1985 he earned Doctor of Science (higher doctorate) with thesis "Computational Complexity in Polynomial Algebra".
Since 1988 till 1992 he was the head of Laboratory of algorithmic methods Leningrad Department of the Steklov Mathematical Institute.
During 1992–1998 Grigoriev hold the position of full professor at Penn State University. Since 1998 he hold the position of Research Director at National Center for Scientific Research, University of Rennes 1, and since 2008 – Research Director at National Center for Scientific Research, Laboratory Paul Painleve University Lille 1 in France.
He is member of editorial boards of the Journal Computational Complexity, Journal of Applicable Algebra in Engineering, Communications and Computations and Groups, Complexity, Cryptology.
Married Alla N. Sokolova, February 21, 1975. 1 child, Andrei D.