This book is designed for use as a text at either the upper-division or graduate level. It is designed for students who have had experience with linear algebra and a solid semester of modern algebra. No previous acquaintance with modules is assumed. To the Student. The approach I have taken to module structure crystallized in the early 1960s. It was made possible by three papers, Eckmann and Schopf [1953J, Matlis [1958J, and Kaplansky [1958J, and the initial sections of a book of mammoth scope, Cartan and Eilenberg's Homological Algebra [1956]. Further results on module structure appear regularly, but I believe that the material presented in this book constitutes a firm central core for the theory of module structure.