This monograph provides a unified theory of maps and their enumerations. The crucial idea is to suitably decompose the given set of maps for extracting a functional equation, in order to have advantages for solving or transforming it into those that can be employed to derive as simple a formula as possible. It is shown that the foundation of the theory is for rooted planar maps, since other kinds of maps including nonrooted (or symmetrical) ones and those on general surfaces have been found to have relationships with particular types in planar cases. A number of functional equations and close formulae are discovered in an exact or asymptotic manner. Audience: This book will be of interest to college teachers, graduate students working in mathematics, especially in combinatorics and graph theory, functional and approximate analysis and algebraic systems.