We formulate the theory of light scattering from polaritons in ionic crystals by a Green's-function method. The theory is presented in a form that allows the effect of the damping of the lattice motion to be included in a rigorous fashion. Difficulties encountered in earlier phenomenological approaches to the problem are eliminated by the present work. It is pointed out that, in principle, detailed studies of the shape of the Raman line may be utilized to determine the frequency dependence of the phonon proper self-energy over a wide range of frequencies. We present the results of numerical studies of the shape and position of the Raman line as a function of scattering angle, for the case where the frequency dependence of the proper self-energy is ignored. We find that even when the damping of the lattice is appreciable, the position of the line center as a function of scattering angle is given accurately by employing the polariton dispersion relation appropriate to the case where no damping is present.