Abstract: The conformal bootstrap is a set of methods to constrain and solve strongly interacting quantum field theories. In this talk, I will describe how we use the bootstrap to study black holes in quantum gravity, and vice-versa. The emphasis is on the role of higher topologies in the gravitational path integral, how they arise from the bootstrap, and what they tell us about quantum field theory. This has applications to ordinary, non-holographic quantum field theories, and to problems in quantum gravity including closed universes and the encoding of the black hole interior.