For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $κ$ such that $P$ is embeddable in a direct product of $κ$ totally ordered sets. We study the behavior of this function on finite-dimensional (not necessarily finite) posets. In general, the dimension dim($P$ x $Q$) of a product of two posets can be smaller than dim($P$) + dim($Q$), though no cases are known where the discrepancy is greater than 2. We obtain a result that gives upper bounds on the dimensions of certain products of posets, including cases where the discrepancy 2 is achieved. But the paper is mainly devoted to stating questions, old and new, about dimensions of product posets, noting implications among their possible answers, and introducing some related concepts that might be helpful in tackling these questions.