During 1976–1992 he was with LOMI, Leningrad Department of the Steklov Mathematical Institute of the USSR Academy of Sciences.

In 1979 he earned PhD (Candidate of Sciences) in Physics and Mathematics with thesis "Multiplicative Complexity of a Family of Bilinear Forms" (from LOMI, under the direction of Anatol Slissenko).

Since 1988 until 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.

[6] He is recipient of the Prize of Leningrad Mathematical Society[7] (1984), Max Planck Research Award of the Max Planck Society, Germany (1994), and Humboldt Prize of Humboldt Foundation, Germany (2002), Invited Speaker of International Congress of Mathematicians, Berkeley, California, 1986.