Recent interest in stochastic traffic assignment models has been motivated by a need to determine the stationary probability distribution of a network's traffic volumes and by the possibility of using time-series of traffic counts to fit and test travel demand models. Because of the way traffic volumes are generated as the sum of path flows from different origin-destination pairs, and because of the nonlinear nature of the process relating traffic conditions to traveler route selection, most plausible assignment models tend to be intractable. In this paper, we first pose a general stochastic assignment model that includes as special cases most models which have appeared in the literature, and then verify that the probability distributions of an equivalent Markovian model converge to a stationary distribution. We next verify that as the number of individual travelers becomes large, the general model can be approximated by the sum of a nonlinear deterministic process and a time-varying linear Gaussian process. The stationary distribution of this approximation is readily characterized, and the approximation also provides a means for employing linear system methods to estimate model parameters from a set of observed traffic counts. For the case where the route choice probabilities are given by the multinomial logit function, computationally feasible procedures for implementing the approximate model exist.