Maps as a mathematical topic arose probably from the four color problem[Bir1, Ore1] and the more general map coloring problem[HiC1, Rin1, Liu11] in the mid of nineteenth century although maps as polyhedra which go back to the Platonic age. I could not list references in detail on them because it is well known for a large range of readers and beyond the scope of this book. Here, I only intend to present a comprehensive theory of maps as a rigorous mathematical concept which has been developed mostly in the last half a century.