We consider completely continuous and weakly compact multiplication operators on certain classical function spaces, more precisely on Lebesgue spaces $L^1$ on spaces $C(K)$ of continuous functions on a compact Hausdorff space K,and on the Hardy space $H^1$. We will describe such operators in terms of their defining symbols. Our characterizations extend corresponding results known from the literature. In any case, our results reveal the severe restrictions on the symbols of multiplication operators necessary to ensure complete continuity or weak compactness. The apparent simplicity of the obtained descriptions belie the deep and beautiful functional analytic principles that underlie them.