  In the past decade there has been a flurry of activity at the intersection of
spectral theory and symplectic geometry. In this paper we review recent results
on semiclassical spectral theory for commuting Berezin-Toeplitz and
h-pseudodifferential operators. The paper emphasizes the interplay between
spectral theory of operators (quantum theory) and symplectic geometry of
Hamiltonians (classical theory), with an eye towards recent developments on the
geometry of finite dimensional integrable systems.
