On a Riemannian manifold a lower bound on Ricci curvature controls the distortion of volume in the space. This principle leads to control on various geometric, analytic, and topological quantities on the manifold. The Bakry-Emery Ricci curvature is generalization of Ricci curvature to the setting of a Riemannian manifold with a smooth measure. In this talk I'll first give a broad overview of some of the basic comparison theorems for Ricci curvature and their topological consequences, then I'll discuss versions of these results for the Bakry-Emery Ricci curvature.