The review of the recent advances made in the western part of the world in the theory of radiative transfer relates to the studies of the only physically realistic models of a planetary atmosphere in which both multiple scattering and polarization are taken fully into consideration. In the mathematical formulation of radiative transfer problems, the meaningful advances have been made in the emphasis on better physical interpretation of the mathematical analysis and of the used parameters that have resulted in the renaissance of the original Chapman's approach, based on a source matrix and on the solution of auxiliary equations for this matrix. This procedure is demonstrated in the derivation of this equation for the inhomogeneous case of imperfect scattering (i.e., the case with absorption and scattering with variable albedo of single scattering within the medium). The integrodifferential equations for the reflection and transmission matrices for this case are presented, and their reduction is demonstrated for the separable phase matrix in which the directional parameters of the incident and scattered radiation can be separated. For such a phase matrix the integrodifferential equations can be reduced to the solution of a pair of integral equations forXandYmatrices that for homogeneous cases assume forms identical to those for theXandYfunctions introduced by Chandrasekhar. Considerable advances have been made in the analysis of these functions, especially in the solution of the problem of their non‐uniqueness. With the use of the theory of singular integral equations applied to the nonlinear integral equation forXandYfunction, transformed into a singular form, a method has been recently developed that enables rapidly converging iteration for large optical thickness, leading even to asymptotic expressions for large τ. The review is concluded with the list of problems that can be expected to be solved in the near future without great or basic difficulti