A Guide to the Classification Theorem for Compact Surfaces The Classification Theorem: Informal Presentation Surfaces Simplices Complexes and Triangulations The Fundamental Group Orientability Homology Groups The Classification Theorem for Compact Surfaces Viewing the Real Projective Plane in R3 Proof of Proposition 5.1 Topological Preliminaries History of the Classification Theorem Every Surface Can be Triangulated Notes . This welcome boon for students of algebraic topology cuts a much needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge or too informal to afford students a comprehensive insight into the subject. Its dedicated student centred approach details a near complete proof of this theorem widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured worked example. Ideal for students whose mastery of algebraic topology may be a work in progress the text introduces key notions such as fundamental groups homology groups and the Euler Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused discrete learning where it is required without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles theory and applications of the classification theorem the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.