A fundamental principle in Riemannian geometry is that bounds on the sectional curvature are equivalent to appropriate convexity properties of the underlying metric space. In this talk we will develop a generalization of this theory to Riemannian manifolds equipped with a density function. We obtain generalizations of many of the classical comparison results such as the (non-smooth) 1/4-pinched sphere theorem, the splitting theorem, and the Cartan-Hadamard theorem. This is joint work with Lee Kennard (Univ. of Oklahoma) and Dmytro Yeroshkin (Idaho State).