If R is a ring with 1, we call a unital left R-module M co-Hopfian (Hopfian) in the category of left R-modules if any monic (epic) R -module endomorphism of M is an automorphism. In the case that R is commutative Noetherian, we use results of Matlis to show that, in a particular setting, every submodule of a co-Hopfian injective module is co-Hopfian. We characterize when a finitely generated co-Hopfian module over a commutative Noetherian ring has finite length. We describe the structure of Hopfian and co-Hopfian abelian groups whose torsion subgroup is cotorsion.