Written for graduate students, and accessible to upper-level undergraduates, this book presents topics in two-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane, which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view: first the density is defined and then the metric via integration.The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains.New material on hyperbolic and hyperbolic-like metrics is presented. These are dynamical generalizations of the Kobayashi and Caratheodory metrics for plane domains. This book concludes with applications to holomorphic dynamics, including new results and accessible open problems.