  In this paper, we study the multiplication operators on the Bloch space of a
bounded homogeneous domain in $\mathbb{C}^n$. Specifically, we characterize the
bounded and the compact multiplication operators, establish estimates on the
operator norm, and determine the spectrum. We prove that the only bounded
multiplication operators on the Bloch space of the polydisk are those whose
symbol is constant. Furthermore, we prove that for a large class of bounded
symmetric domains, the isometric multiplication operators are those whose
symbol is a constant of modulus one.
