First I will review the basic concepts in symplectic dynamics. Then I will describe some recent results concerning classical and quantum integrable systems, emphasizing the interplay between symplectic geometry and spectral theory/semiclassical analysis. I will also briefly describe some random counterparts of classical results in symplectic geometry.

This talk will assume no background on symplectic geometry or semiclassical analysis. The talk is based on joint works with Laurent Charles, Leonid Polterovich, Fraydoun Rezakhanlou, and San Vu Ngoc.