Monte Carlo calculations based on the adjoint transport equation offer an attractive alternative to calculations based on the transport equation when the detector region is much smaller than the source region. However, when an analog simulation of the adjoint equation is attempted, extra variance may arise due essentially to the nonphysical aspects of the adjoint equation. In this paper, a new adjoint Monte Carlo technique is described in which most of this additional variance has been eliminated. The method appears to be very useful for solving slowing down problems involving energies below the threshold for inelastic scattering. The basis for the technique is the idea of exactly reversing direct Monte Carlo random walks. It is shown that this reversal may be accomplished via a transformation of the adjoint transport equation by means of a discontinuous importance function. This transformation is a logical extension to continuous energies of an adjoint multigroup formulation used by Gelbard and Spanier to study thermal problems. Numerical results are provided which illustrate the variance reduction resulting from the use of this technique.