We develop the theory of module categories over a Grothendieck-Verdier category C, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category comes with two monoidal structures ⊗ and [Figure presented] which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category M we find two distinguished subcategories [Figure presented] and Mˆ⊗, which can be characterized by certain structure morphisms being actually invertible. The internal Hom Am:=Hom_(m,m) of an object m in Mˆ⊗ that is a C-generator is an algebra such that mod-Am is equivalent to M as a module category. Crucially, the subcategories [Figure presented] and Mˆ⊗ are precisely those on which a relative Serre functor can be defined. This relative Serre functor furnishes an equivalence [Figure presented], and any isomorphism m→≅S(m) endows the algebra Am with the structure of a Grothendieck-Verdier Frobenius algebra.