The classification theorem for compact surfaces is one of the great achievements of early 20th century mathematics. The statement of this theorem is quite intuitive but it took about sixty years until a rigorous proof was finally given by Brahana in 1921. Early versions of the classification theorem were given by Mobius in 1861, and by Jordan in 1866. More definite proofs were given later by von Dyck in 1888 and Dehn and Heegaard in 1907.

This talk is about the history of the theorem and the techniques used to prove it. We will give a guided tour of the proof, pointing out which tools from algebraic topology are needed, and give an abbreviated history of the "proof." A byproduct of the theorem yields ``global parametrizations," using fundamental domains, a recent topic of research.