The cyclic behavior of a composition operator is closely tied to the dynamical behavior of its inducing map. Based on analysis of fixed-point and orbital properties of inducing maps, Bourdon and Shapiro show that composition operators exhibit strikingly diverse types of cyclic behavior. The authors connect this behavior with classical problems involving polynomial approximation and analytic functional equations. Features include: complete classification of the cyclic behavior of composition operators induced by linear-fractional mappings; application of linear-fractional models to obtain more general cyclicity results; and, information concerning the properties of solutions to Schroeder's and Abel's functional equations. This pioneering work forges new links between classical function theory and operator theory, and contributes new results to the study of classical analytic functional equations.