Particle transport in rod and plane geometry random media is considered. The cross section is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. In rod geometry, an exact closure is constructed for semi-infinite media that yields exact equations for the ensemble-averaged scalar flux Φ and current J. The same closure scheme yields a Fokker-Planck equation for the joint probability distribution function of Φ and J, from which ensemble-averaged equations for higher order quantities are derived and solved exactly for an arbitrary correlation function. Finally, the penetration of a beam of charged particles in a highly forward scattering random medium is considered, and circumstances that yield a closed ensemble-averaged transport equation are determined.