Measurements of the polarization of light scattered from the beam of a helium-neon gas laser at low pressures are described. The intensity, polarization and parallelism of the beam permit high accuracy, and new values for the depolarization ratios of twenty-four simple species are reported. The general quantum theory of scattering is discussed and applied in detail to the evaluation of a formula for the depolarization ratio of the scattered light. It is found that quantum corrections to the classical formula arise from (i) the effects of frequency changes due to rotational Raman scattering, (ii) changes in the molecular polarizability with rotational state due to centrifugal distortion, (iii) approximations inherent in the polarizability scattering formula, and (iv) vibrational Raman scattering. Effect (i) reduces the depolarization of hydrogen to 91% of its classical value; (iii) is unimportant unless the frequency of the light is near an absorption frequency of the molecule. The depolarization measurements have been combined with refractivity data to yield the anisotropies in molecular polarizabilities of the molecules studied.