As an introductory book, this book contains the elementary materials in map theory, including embeddings of a graph, abstract maps, duality, orientable and non-orientable maps, isomorphisms of maps and the enumeration of rooted or unrooted maps, particularly, the joint tree representation of an embedding of a graph on two dimensional manifolds, which enables one to make the complication much simpler on map enumeration. All of these are valuable for researchers and students in combinatorics, graphs and low dimensional topology.

A Smarandache system (Sigma;R) is such a system with at least one Smarandachely denied rule, non-r in R, such that it behaves in at least two different ways within the same set Sigma, i.e. validated and invalided, or only invalided but in multiple distinct ways. A map is a 2-cell decomposition of surface, which can be seen as a connected graph in development from partition to permutation, also a basis for constructing Smarandache systems, particularly, Smarandache 2-manifolds for Smarandache geometry.