  A longest sequence $S$ of distinct vertices of a graph $G$ such that each
vertex of $S$ dominates some vertex that is not dominated by its preceding
vertices, is called a Grundy dominating sequence; the length of $S$ is the
Grundy domination number of $G$. In this paper we study the Grundy domination
number in the four standard graph products: the Cartesian, the lexicographic,
the direct, and the strong product. For each of the products we present a lower
bound for the Grundy domination number which turns out to be exact for the
lexicographic product and is conjectured to be exact for the strong product. In
most of the cases exact Grundy domination numbers are determined for products
of paths and/or cycles.
