Recent advances in the theory of complex Monge-Amp\`ere equations, and more generally, fully non-linear equations, have opened up new avenues for the study of geometric invariants such as the diameter, the localized volume form, and the Green’s function in K\”ahler geometry. In particular, the frequent assumptions on the Ricci curvature in classical results can now be weakened to integral bounds on the volume form. We describe these advances and their applications to the K\”ahler-Ricci flow. This is joint work with B. Guo, J. Song, and J. Sturm.