Burnside's Theorem states that if the order of a finite group has only two different prime divisors, then the group is solvable. The original proof is an amusing application of results from the representation theory of finite groups. We shall discuss Maschke's theorem, the Schur orthogonality relations and some properties of algebraic integers, and then present a proof of Burnside's Theorem. Prerequisites will be kept to a minimum; some familiarity with modules might be useful.