This book provides a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years. More recently, many authors (including the authors of this book) have investigated relationships between categories of modules over a pair of rings that are induced by both covariant and contravariant representable functors, in particular, by tilting and cotilting theories. Collecting and unifying the basic results of these investigations with innovative and easily understandable proofs, the authors provide an aid to further research on this central topic in abstract algebra.