This monograph provides a thorough treatment of two important parts of contemporary module theory: approximations of modules and their applications, notably to infinite dimensional tilting theory, and realizations of algebras as endomorphism algebras of groups and modules. Attention is also given to E-rings and free modules with distinguished submodules. The monograph starts from basic facts and gradually develops the theory to its present frontiers. It is suitable for graduate students interested in algebra as well as experts in module and representation theory.