  We provide necessary and sufficient conditions for the generalized
$\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one
solution for any right-hand side E. These conditions are given for arbitrary
coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize
existing results for the same equation with square coefficients. We also review
the known results regarding the existence and uniqueness of solution for
generalized Sylvester and $\star$-Sylvester equations.
