Tlhe first section of this paper will be devoted to the study of two particular surfaces. The methods used to treat thiese surfaces will then be extended to develop a classificationof surfaces, each class consisting of combinatorially eauivalent surfaces. Applications will be made to tlhe study of the topological nature of covering surfaces and of Riemann surfaces in particular. It will be assumed that the reader is familiar with the material of Chapter V, pages 235-244 and 256-264, in the book What Is 4lathsrmatics? by Courant and Robbins. Other than for references to this book, no previous knowledge of Topology will be assumed.