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.