An important class of mathematical resonance problems arises for Hamiltonian partial differential equations, which may be viewed as consisting of two coupled subsystems: a finite dimensional part governing "oscillators" with discrete frequencies and an infinite dimensional part, governing "waves" with a continuous spectrum of frequencies. We first discuss several examples and then describe work on ground state selection and energy equi-partition for nonlinear Schroedinger / Gross-Pitaevskii equations. Finally, we discuss confirmation of predictions in nonlinear optical experiments.