In this chapter (and the next) weFinite noncommutative space consider finite discrete topological spaces. However, we will stretch their usual definition, which is perhaps geometrically not so interesting, to include the more intriguing finite noncommutative spaces. Intuitively, this means that each point has some internal structure, described by a particular noncommutative algebraAlgebra. With such a notion of finite noncommutative spacesFinite noncommutative space, we search for the appropriate notion of maps between, and (geo)metric structure on such spaces, and arrive at a diagrammatic classification of such finite noncommutative geometric spaces.