the inverse problem in spectral theory was proposed by bochner and weyl in the early 20th century. one formulation of the question is: how much of the dynamics of a classical dynamical system can be detected from the spectrum of its quantization? i will describe this question and review recent results for the case integrable dynamical system, where in certain fundamental cases going back to the work of atiyah and guillemin-sternberg a full solution can be given. the talk will emphasize the interplay between symplectic geometry and semiclassical spectral theory and is intended for a general audience.