The Golod-Shafarevich theorem gives a sufficient condition for an associative algebra presented by generators and relators to be infinite dimensional. Immediately upon arrival in 1964, this result had important consequences in various fields of mathematics. Golod used the theorem to construct a finitely generated, infinite, torsion group--the first counterexample to the general Burnside problem. Shafarevich used it to construct the first infinite tower of class fields. The primary topics of this talk will be the theorem, the Burnside problem and Golod's example.