This book, a first comprehensive review on statistical spectroscopy, deals with two related yet distinct topics — averages and fluctuations. While fluctuations have been dealt with in considerable detail in Porter's book entitled Statistical Theories of Spectra: Fluctuations, and subsequent reviews and books, there does not exist at present a similar treatise on averages. This unique volume is designed to fill this significant gap.The book begins with an introductory review and overview of the subject of spectral distributions initiated by J Bruce French in the 60's followed by a collection of original papers which continue to give new insight on average properties of spectra. The purpose is to highlight the considerable advancements made in the application of statistical spectroscopy to nuclear structure and to encourage new directions in random matrix theory, many-body chaos and statistical mechanics of finite quantum systems such as nuclei, atoms, molecules, quantum dots, etc.Along with Wong's book entitled Nuclear Statistical Spectroscopy which, though not a review, deals with most of the important techniques, this volume would be useful to one looking for a thorough introduction to the subject as well as to the specialist contemplating new applications. Finally, with most of the material available in one place, this book would be ideal in the design of graduate courses in statistical spectroscopy suited to specific needs.