We give an introduction on discrete Ricci curvature notions and give an overview of recent results. In particular, we focus on Ollivier Ricci curvature which has been introduced via optimal transport theory. A characterization of lower Ricci curvature bounds via gradient estimates for the heat semigroup is presented. We show that non-negative Ricci curvature implies the Liouville property, i.e., every bounded harmonic function is constant. This seems to be the first analytic result for graphs with non-negative Ricci curvature in the sense of Ollivier.