This text deals with classical and contemporary topics in function theory and is designed to be used after a one-year course in real and complex analysis. It can be used as a text for topics courses or courses in function theory, operator theory and applied areas. The first six chapters supplement the authors' book "Hardy Classes and Operator Theory". The theory of harmonic majorants for subharmonic functions is used to introduce Hardy-Orlicz classes, which are specialized to standard Hardy classes on the unit disk. The theorem of Szegoe-Solomentsev characterizes boundary behaviour. Half-plane function theory receives equal treatment and features the theorem of Flett and Kuran on existence of harmonic majorants and applications of the Phragmen-Lindeloef principle. The last three chapters contain an introduction to univalent functions, leading to a self-contained account of Loewner's differential equation and de Branges' proof of the Milin conjecture.