The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This second edition of Hyperbolic Geometry has been thoroughly rewritten and updated. Chapter 4 focuses on planar models of hyperbolic plane that arise from complex analysis and looks at the connections between planar hyperbolic geometry and complex analysis.However most of the new material will appear in Chapter 6 and concentrates on an introduction to the hyperboloid model of the hyperbolic plane. The chapter concludes with a discussion of hyperbolic geometry in higher dimensions, and generalizations of hyperbolicity (this, in particular, is an important topic that allows for an in-depth development of the fundamental concepts). This book is written primarily for third or fourth year undergraduate students with some calculus knowledge. It contains new exercises with solutions and is ideal for self-study or as a classroom text.