  We prove that rationally essential manifolds with suitably large fundamental
groups do not admit any maps of non-zero degree from products of closed
manifolds of positive dimension. Particular examples include all manifolds of
non-positive sectional curvature of rank one and all irreducible locally
symmetric spaces of non-compact type. For closed manifolds from certain
classes, say non-positively curved ones, or certain surface bundles over
surfaces, we show that they do admit maps of non-zero degree from non-trivial
products if and only if they are virtually diffeomorphic to products.
