Abstract: The complex of curves of a surface is a complex that encodes information about simple closed curves. The vertex set corresponds to isotopy classes of simple closed curves and disjointness determines if there is an edge between vertices. For low complexity surfaces the curve complex is simple to describe, but for higher complexity the curve complex is locally infinite and has infinite diameter. The mapping class group of the surface naturally acts on the curve complex and this action has been used extensively to prove many properties about the mapping class group. In this talk, we will define the curve complex, look over some basic examples, and explore some results and applications of the complex. This talk is meant to be introductory and should be accessible to anyone.