Abstract. The familiar Brouwer fixed point theorem says that any continuous self­map f on a compact convex subset X of finite dimensional Euclidean space E must leave at least one point fixed. This result is easy to state, but notoriously complicated to prove. We will give a sample of the various methods of proof available, ranging from the degree­theoretical methods used by Brouwer in the early 20th century, up to a recent proof based on an alternate change of variables formula for multiple integrals. We will also explore extensions of the theorem based on generalizations the space E, the set X, and the function f .