The Monte Carlo approach to the solution of radiant energy transfer among surfaces of a system represents radiant energy flux statistically as rays which can be traced through the system according to a set of probability distribution functions. Although the ray repre-sentation of the radiant energy is similar to that of a photon characterization, it is treated in this method as merely a statistical representation. Previous computer programs of this type have experienced considerable problems in achieving attenuation prediction accuracy, with acceptable confidence, in a reasonable computer run time. Traditionally, problems have arisen in the areas of excessive computer time requirements, poor statistical convergence, biased results, and related problems. Advanced Monte Carlo variance reduction techniques were applied in the development of the GUERAP III program which have alleviated problems in this area. The techniques of ray-splitting, expected value, and importance sampling have led to very significant reductions in the required GUERAP run times.