Group rings, RG, and their modules play a central role in the structure theory of groups. The present monograph concentrates on the special case of artinian modules over the RG's, where R is a Dedekind domain and G is an infinite group satisfying some finiteness condition. The monograph consists of seventeen chapters. After several preliminary chapters on groups and modules, the Kovacs-Newman generalization of the Maschke theorem is presented in chapter 7 and Hartley's characterization of the V-rings of form RG where R is a field and G is a countable group in chapter 9. The core of the book naturally covers results on artinian modules: their Baer decompositions for G locally soluble RG-hypercentral obtained in chapter 10 and their structure for G abelian of finite section rank in chapter 14. In the final chapter, as an application of the tools developed, the authors present a new proof of Robinson's theorem concerning splitting of a group over its abelian generalized nilpotent radical. Equipped with a list of 313 references, the book will be useful for students and researchers both in group theory and in ring and module theory.