Abstract: The black hole information paradox, discovered by Hawking in 1976, is that black hole evaporation seems to violate quantum mechanics by creating too much entropy. I will describe the current status of this problem and discuss recent progress, including a new calculation of the entropy of Hawking radiation that is compatible with unitary quantum mechanics. There are two key ingredients: a link between classical geometry and quantum entanglement, and higher topologies in quantum gravity known as replica wormholes. I will also discuss the prospects for applying similar methods to cosmology in the early universe, which is similar in some ways to the inside of a black hole.