Purpose The major topic in this book is spectral synthesis. The purpose of the book can be described as follows: A. To trace the development of spectral synthesis from its origins in the study of Tauberian theorems; B. To draw attention to other mathematical areas which are related to spectral syn- thesis; C. To give a thorough (although not encyclopedic) treatment of spectral synthesis for 1 the case of L ( G); D. To introduce the integration and structure problems that have emerged because of the study of spectral synthesis. A and B. The first two points are discussed in Chapters 1 and 2, and are the major reasons that such a large bibliography has evolved. By the end of Chapter 2, the significant relationship between Tauberian theorems and spectral synthesis is not only firmly established, but the extent to which this relationship is still undetermined is emphasized by the open C-set-S-set problem. Also, Chapters 1 and 2 view spectral synthesis amidst other problems and influences. C. By contrast, Chapter 3 is (or at least is meant to be) more business-like, and synthesis is essentially the only topic under discussion.