We are concerned with localization properties of solutions to hy- perbolic PDEs, especially problems with a geometric component: how do boundaries affect spreading and concentration of solutions. While our first focus is on wave and Schrodinger equations on manifolds with boundary, strong connexions exist with phase space localization for eigenfunctions, which are of independent interest. Motivations come from nonlinear dispersive models, properties of eigenfunctions in quantum chaos or harmonic analysis on manifolds.