A major theme of geometric group theory is the topological and geometric properties of a space that a specific group acts on. The study of the mapping class group associated to a surface of finite-type led to various descriptions and classifications using the set of simple closed curves. In this talk I will introduce the curve complex and discuss various interesting properties. The curve complex of the torus is a special case that can be given an explicit treatment using continued fractions. The higher complexity cases are related to Teichmüller theory and one way to see it is through the shadows of Teichmüller geodesics. The curve complex is hyperbolic and studying the Gromov boundary of the curve complex will give us an interesting result about cylinders and flat surfaces. The conclusion of this talk is joint work with Chris Leininger.