This book is intended as an (advanced) introductory text on the main results and ideas in some of the major topics of modern operator theory. The first half of the book is devoted to the study of completely positive and completely bounded maps between C*-algebras and their connections with the dilation theory. The adverb completely is related to a collection of matrix norms on the algebra. The second part explains connections with various types of similarity problems (e.g. the Kadison conjecture and Pisier's theory), and properties of operator systems (e.g., characterization of existence of isometric representations of operator algebras - Blecher, Ruan, Sinclair theory, which is applied to new proofs of classical results on interpolation of analytic functions). Another application is Pisier's theory of the universal operator algebra of an operator space and his results on similarity and factorization degree. The book is carefully written, proofs are often accompanied with notes helping to explain the situation. Several theorems are proved by various methods, e.g., there are five different proofs of the von Neumann inequality. The text can be used either for a graduate course in operator theory or for an independent study, since it contains 205 exercises. The author recommends that the reader be acquainted with R. G. Douglas' book “Banach Algebra Techniques in Operator Theory”.