During the Fall of 1982 and Spring of 1983, Andrew Casson gave a course in surface automorphisms at the University of Texas. This book began as my lecture notes from that course. Andrew and I have revised these notes into the present form. With this book we hope not only to give students an introduction to hyperbolic geometry and its applications to low–dimensional topology, but also provide a concise exposition of the Nielsen–Thurston theory of surface automorphisms. The reader should have an understanding of the basic lecture courses in topology and linear algebra. We begin by recalling the structure of the self–homeomorphisms of the torus.