Let R be a ring with an identity 1. Let A, B, and C be R-modules. The sequence A → [f above arrow] B→[g above arrow] C is exact providing f and g are R-homomorphisms and Im f =Ker g. Let 0 represent the R-module with precisely one element. An R-module J is injective if and only if for every exact sequence 0→A→ [f above arrow] B of R-modules and R-homomorphisms and every R-homomorphism g: A→J there exists an R-homomorphism h: B→J such that hf = g. This is a dual concept to that of a projective R-module. In the second chapter the idea of an injective R-module is studied quite intensively, and several different characterizations of injective · modules are proved. One of the principal results obtained is that every R-module is a submodule of an injective R-module. Further properties of injective R-modules are given in Chapter 3, including the concepts of injective dimension and an injective resolution of an R-module. Using these concepts the Shifting Theorem for injectives is proved. The basic definitions and results necessary for the development of the concept of injective for abstract categories are included in Chapter 4. An injective object is then defined in this general setting. Then the concept of an injective envelope is defined. The problems that arise, in the effort to restrict the category of topological groups to the appropriate subcategory so that the concept of an injective topological group is of interest, are investigated in Chapter 5. The development of the concept for one such restriction concludes this thesis. / Master of Science