  In this paper we study Cartesian products of graphs and their divisorial
gonality, which is a tropical version of the gonality of an algebraic curve. We
present an upper bound on the gonality of the Cartesian product of any two
graphs, and provide instances where this bound holds with equality, including
for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$. We use our upper
bound to prove that Baker's gonality conjecture holds for the Cartesian product
of any two graphs with two or more vertices each, and we determine precisely
which nontrivial product graphs have gonality equal to Baker's conjectural
upper bound.
