In this thesis we investigate the existence of superdecomposable modules over integral domains, i.e. of modules which do not have any non-zero indecomposable direct summands. We generalize a result of Benabdallah and Birtz {1} about superdecomposable abelian groups and present constructions of superdecomposable modules over noetherian domains, generalized Krull domains, certain valuation domains, and h-local domains. We classify the Dedekind domains and Krull domains of characteristic zero which admit superdecomposable modules
