This book studies the original results, and their extensions, of the Russian mathematician, S.A.Gerschgorin, who wrote a seminal paper, in 1931, on how to easily obtain estimates of all n eigenvalues (characteristic values) of any given n-by-n complex matrix. Since the publication of this paper, there hae been many newer results spawned by his paper, and this book will be the first which is devoted solely to this resulting area.As such, it will include the latest research results, such as Brauer ovals of Cassini and Brualdi lemniscates, and their comparisons. This book is dedicated to the late Olga Taussky-Todd and her husband, John Todd. It was Olga who brought to light Gerschgorin's paper and its significance to the mathematical world. The level of this book requires only a modest background in linear algebra and analysis, and is therefore comprehensible to upper -level and graduate level students in mathematics.