Dielectric cavities can support long-lived (high-Q) resonant states of the electromagnetic field. These resonances correspond to ray trajectories which are trapped inside the cavity by total internal reflection. Typical examples are the "whispering gallery" resonances of microspheres and microcylinders. When such cavities are deformed from their symmetric shapes, the dynamics of the corresponding ray trajectories undergoes a transition from integrability to chaos. This transition has dramatic effect on the properties of the high-Q resonances. Novel microcylinder lasers based on this effect have recently been demonstrated with highly directional emission and peak power three orders of magnitude higher than previous micro-disk lasers.