Spherical and cylindrical dielectric cavities support high-Q modes due to total internal reflection of the trapped light. These modes are analogous to the well-known whispering gallery modes of acoustics. When such cavities undergo smooth convex deformations, the ray dynamics corresponding to these modes becomes chaotic in a universal manner determined by the Kolmogorov--Arnold--Moser theory of classical Hamiltonian mechanics. Using concepts from the theory of wave/quantum chaos, it is argued that the corresponding resonances broaden and become highly anisotropic in a universal manner, independent of the wavelength. Comparison with exact numerical results [J. U. Noeckel and A. Douglas Stone, Nature 385, 45 (1997)] shows that the predicted behavior describes the emission anisotropy extremely well, but the resonance lifetime has important corrections which are conjectured to arise due to the dynamical localization of photons and to chaos-assisted tunneling. The theory is able to explain the lasing intensity profile observed in microdroplets and related experiments, and may eventually have applications to microlasers and in fiber-optic communications networks.