This book deals with three types of operators on Bergman and Hardy spaces on the open unit disk in the complex plane: Töplitz operators, Hankel operators and composition operators. The main emphasis is on the size of these operators, or more precisely whether they are bounded, compact or belong to Schatten classes. The book starts with a presentation of types of operators on Banach and Hilbert spaces that are studied later along with their basic properties. Then the author proves the classical interpolation theorems and Hölder type inequalities on Lp spaces. After introducing Bergman, Bloch and Besov spaces, the author presents results on the Berezin transform. The next sections are devoted to a study of Töplitz and Hankel operators on Bergman spaces. The presentation proceeds with Hankel operators on Hardy and BMO spaces. The last chapter is devoted to composition operators. The second edition is considerably improved and enriched with recent results. Also, several new proofs are included. The book contains exercises of varying degrees of difficulty and an extensive bibliography.