Abstract: Galois cohomology is a topic that should interest a wide range of audiences: number theorists, algebraic geometers, homotopy theorists, etc. It is a wonderful example of an application of ideas from algebraic topology to study algebra and number theory. In this talk, we will discuss the basics of Galois cohomology, and we demonstrate its power by using it to prove the Mordell-Weil theorem for elliptic curves. If time permits, we might also discuss a little bit about its generalization: étale cohomology.