The Riemann Hypothesis is probably the most famous unresolved problem in all of mathematics. While some unsolved problems in math can be stated and understood in a handful of lines, the Riemann Hypothesis requires a significant background before the problem can be grasped. This paper provides the background necessary to understand the Prime Number Theorem, and the zeta function on which the Riemann Hypothesis is based. This paper also examines the tie between the zeta function and the distribution of prime numbers. Operations of complex numbers are discussed at length, as well as functional convergence with complex arguments. The difficulty of graphing functions with complex arguments is addressed and primitive graphs of the zeta function are explained. Lastly, this thesis discusses why this problem has become so important and the current state of the search for its proof.