The subject of latin squares is an old one and it abounds with unsolved problems, many of them up to 200 years old. In the recent past one of the classical problems, the famous conjecture of Euler, has been disproved by Bose, Parker, and Shrikhande. It has hitherto been very difficult to collect all the literature on any given problem since, of course, the papers are widely scattered. This book is the first attempt at an exhaustive study of the subject. It contains some new material due to the authors (in particular, in chapters 3 and 7) and a very large number of the results appear in book form for the first time. Both the combinatorial and the algebraic features of the subject are stressed and also the applications to Statistics and Information Theory are emphasized. Thus, I hope that the book will have an appeal to a very wide audience. Many unsolved problems are stated, some classical, some due to the authors, and even some proposed by the writer of this foreword. I hope that, as a result of the publication of this book, some of the problems will become theorems of Mr. So and So.