We compare and relate two main approaches to defining Ricci curvature bounds for Riemannian manifolds with metrics of regularity below $C^2$. On the one hand, weak derivatives and the notion of positive distributions can be applied to metrics of low regularity. On the other hand, optimal transport theory gives a characterization of Ricci bounds via displacement convexity of an entropy functional, which applies even to the general setting of metric measure spaces. Both approaches are compatible with the classical definition via the Ricci tensor (and hence with each other) in the case of a smooth metric. We show that distributional bounds imply entropy bounds for $C^1$ metrics and that the converse is true for metrics of regularity $C^{1,1}$ under an additional convergence condition on regularizations of the metric. This is joint work with M. Oberguggenberger and J.A. Vickers.