  We develop the theory of module categories over a Grothendieck-Verdier
category, i.e. a monoidal category with a dualizing object and hence a duality
structure more general than rigidity. Such a category C comes with two monoidal
structures 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 important
subcategories M' and M''. The internal End of an object in M' that is a
C-generator is an algebra such that its category of modules is equivalent to M
as a module category. We also introduce a partially defined relative Serre
functor S which furnishes an equivalence between M' and M''. Any isomorphism
between an object m of M' and S(m) in M'' endows the internal End of m with the
structure of a Grothendieck-Verdier Frobenius algebra.
