Recently powerful techniques have emerged for performing multi-loop computations of scattering amplitudes in quantum gravity and supergravity. These techniques include generalized unitarity and the double-copy property, related to color-kinematics duality in gauge theory. Using these techniques, the ultraviolet divergence structure of N=8 supergravity, and more recently pure N=4 supergravity, have been assessed, not only in four space-time dimensions but also in higher dimensions. The results can be compared to expectations based on potential counterterms that can be constructed using (conjectured) superspace formalisms or nonlinear symmetry constraints. Interestingly, the critical ultraviolet dimension in which N=8 supergravity first diverges is equal to that for N=4 super-Yang-Mills theory through four loops. If this statement were to hold to all loop orders, then N=8 supergravity would represent a perturbatively finite, point-like theory of quantum gravity in four dimensions. In this talk, I will review all of this recent progress.