The use of the Monte Carlo method for the study of deep penetration of radiation into and through shields entails the use of sophisticated methods of variance reduction to make such calculations economical or even feasible. This paper presents an exposition of the most useful methods of variance reduction. The exposition is unified by consistent exploitation of adjoint formulations to estimate expected values, as in previous work, and further to evaluate the variance of the resulting estimates., The connection between adjoint formulations and the choice of biasing schemes is also investigated. In particular, it is shown that the value function (the solution of the integral equation of the adjoint formulation) is always a good choice for importance function biasing; a sharp upper bound, independent of the particular problem, is found for the resulting variance. Predicted (analytic) and experimental (Monte Carlo) results are also given for a simple one-dimensional problem.