The Brouwer fixed point theorem is an important result for proving existence in a wide variety of situations. While there exist many proofs, Brouwer's original proof was based on the notion of degree for continuous maps. As demonstrated by Leray and Schauder in the 1930s, this idea can be generalized from R^n to infinite-dimensional spaces. In this talk, we will explore how Leray and Schauder's theory is used to prove fixed point theorems, and how fixed point theorems are used to prove existence theorems.