The purpose in writing this expository monograph has been three-fold. First, the author set out to present the solution of a problem posed by Wolfgang krull in 1932. He asked whether what is now called the "Krull-Schmidt Theorem" holds for artinian modules. Secondly, the author presents the answer to a question posed by Warfield in 1975, namely, whether the Krull-Schmidt Theorem holds for serial modules. The authors third incentive focuses on the fact that the Krull-Schmidt theorem holds for some, not all, classes of modules. Apart form these issues, the book addresses various topics in module theory and ring theory, some now considered classical and others more specialized. Open problems conclude the work.