The geometrization program of Thurston exposed hyperbolic 3-manifolds to be the least-understood and having the richest structure of Thurston's eight geometries. Since then, there have been many approaches to studying the topology of hyperbolic 3-manifolds, one of which is studying the character variety. This is an algebraic object discovered by Marc Culler and Peter Shalen that contains a wealth of information about the topology of the 3-manifold. In this talk, I will go through Culler and Shalen's original paper establishing the character variety of a hyperbolic 3-manifold as a fundamental tool in understanding its topology. If time permits I will provide examples of its wide-reaching applications in hyperbolic 3-manifolds.