Low-energy quantum gravity has recently shown itself to be surprisingly powerful. Gravity provides geometric descriptions of quantum phenomena, with no reference to a UV completion such as string theory. In particular, the information content of an evaporating black hole and its Hawking radiation can be calculated, shedding new light on the black hole information problem. A common theme in these results is the presence of “spacetime wormholes”: connections between otherwise-distant points of spacetime. But such geometries also appear to give rise to classical randomness in the predictions of the theory. I will explain how these ideas fit together and suggest a new paradigm for understanding quantum gravity.