Abstract: These are lecture notes comprising about half of a short course I gave at the University of Padua during the Summer of 1998. The initial setting is the space H(U) of all holomorphic functions on the unit disc U. In this context the notes discuss of invertibility of composition operators, leading to the classification of conformal automorphisms of the unit disc, and eigenvalues, leading to a discussion of Koenigs's work on Schroeder's functional equation. Then the scene shifts to the Hardy space H^2, with a proof of Littlewood's theorem on boundedness of composition operators, and an introductory discussion of compactness.
These notes can serve as a gentle introduction to the material in my book, Composition Operators and Classical Function Theory, which is itself a gentle introduction to the whole subject of composition operators.