Explore the central theorem in algebraic topology - the classification of connected compact combinatorial surfaces - in this 51-minute lecture from the Insights into Mathematics series. Delve into the introduction of this fundamental result and examine the strategy behind its traditional proof. Learn about key concepts such as polygons, vertices, traversing, Euler number, torus, orientable and non-orientable surfaces, and combinatorial representation. Gain a deeper understanding of the main theorem and its significance in the field of algebraic topology as part of this beginner-friendly series presented by N Wildberger at UNSW.