  Let G be a group acting faithfully on a set X. The distinguishing number of
the action of G on X is the smallest number of colors such that there exists a
coloring of X where no nontrivial group element induces a color-preserving
permutation of X. In this paper, we consider the distinguishing number of two
important product actions, the wreath product and the direct product. Given
groups G and H acting on sets X and Y respectively, we characterize the
distinguishing number of the wreath product of G and H in terms of the number
of distinguishing colorings of X with respect to G and the distinguishing
number of the action of H on Y. We also prove a recursive formula for the
distinguishing number of the action of the Cartesian product of two symmetric
groups S_m x S_n on [m] x [n].
