Almost everything that happens in classical mechanics also shows up in quantum mechanics when we know where to look for it. A phenomenon in classical mechanics involves topological changes in action-angle loops as a result of passage around a "monodromy circuit." This phenomenon is known by the short name "Hamiltonian monodromy" (or, more ponderously, "nontrivial monodromy of action and angle variables in integrable Hamiltonian systems"). In this paper, we show a corresponding change in quantum wave functions: These wave functions change their topological structure in the same way that the corresponding classical action-angle loops change.