Abstract: We discover a wide range of new nonperturbative effects in quantum gravity, namely moduli spaces of constrained saddles (often called constrained instantons) of the Einstein-Hilbert action. We find these in all spacetime dimensions, for AdS and dS. Many can be written in closed form, and some are shown to be quadratically stable. In the Euclidean AdS setting, these constrained saddles correspond to Euclidean wormholes. In 3D AdS, where the gravitational path integral is more tractable, we study these Euclidean wormholes and show that they encode the energy level statistics of microstates of BTZ black holes. Furthermore, these statistics precisely agree with a quantitative prediction from random matrix theory.