Recent years have seen enormous progress in our ability to compute elementary-particle scattering processes in quantum field theory. We will describe the application of modern ideas, including the duality between color and kinematics and advanced loop-integration methods, to studies of the ultraviolet properties of quantum gravity theories. We also describe "enhanced ultraviolet cancellations" for which no standard symmetry explanation has been found, and present the nontrivial example of N = 5 supergravity at four loops. We also show recent results for N = 8 supergravity at five loops, proving that although it is still remarkably well behaved, its behavior is worse than that of N = 4 super-Yang-Mills theory. New structures that should enormously simplify future computations at six loops and beyond will be described. We will also outline recent progress towards applying some of the ideas used in multiloop quantum gravity calculations to the problem of computing classical gravitational radiation from astronomical sources.