Similarity Problems and Completely Bounded Maps This book is mainly about 3 similarity problems arising in 3 different contexts namely group representations C* algebras and uniform algebras (eg. the disc algebra). These 3 problems (all still open in full generality) are studied using a common tool completely bounded maps which have recently emerged as a major concept in operator algebra theory. The book is devoted to the background necessary to understand these problems to the partial solutions that are known and to numerous related concepts results counterexamples or extensions. The variety of topics involved ranging from functional analysis to harmonic analysis Hp spaces Fourier multipliers Schur multipliers coefficients of group representations group algebras characterizations of amenable groups nuclear C* algebras Hankel operators etc is an attraction of this book. It is mostly self contained and accessible to graduate students mastering basic functional and harmonic analysis. For more advanced readers it can be an invitation to the recently developed theory of operator spaces for which completely bounded maps are the fundamental morphisms