The first of two planned volumes on Banach-space isometries, this text characterizes isometries and describes them in classical function spaces. Fleming (mathematics, Central Michigan U.) and Jamison (mathematical sciences, U. of Memphis) give historically important results, expose methods of attack, and include recent and lesser-known results. Topics include Eilenberg's theorem, Lamperti's results, Bergman spaces, Zaidenberg's generalization, and Musielak-Orlicz Spaces.