"Similarity Problems and Completely Bounded Maps" is a specialized book that addresses three central similarity problems that arise in different contexts, all of which pertain to the space B(H) of bounded operators on a complex Hilbert space H. The problems are rooted in the fields of group representations, C*-algebras, and the disc algebra. The book provides a comprehensive introduction to the necessary foundations for understanding these problems and illuminates the known solutions in specific cases, as well as numerous related concepts, results, counterexamples, and extensions that have emerged from the investigation of these issues. A central concept discussed in this context is complete boundedness, which allows for the unification of the three problems within a common framework. The question of whether boundedness implies complete boundedness for linear maps is explored in detail.