This book is about orthomorphisms and complete mappings ofgroups, and related constructions of orthogonal latinsquares. It brings together, for the first time in bookform, many of the results in this area. The aim of this bookis to lay the foundations for a theory of orthomorphismgraphsof groups, and to encourage research in this area. Tothis end, many directions for future research are suggested.The material in this book should be accessible to anygraduate student who has taken courses in algebra (grouptheory and field theory). It will mainly be useful inresearch on combinatorial design theory, group theory andfield theory.