A number of mobility models have been proposed for the purpose of either analyzing or simulating the movement of users in a mobile wireless network. Two of the more popular are the random waypoint and the random direction models. The random waypoint model is physically appealing but difficult to understand. Although the random direction model is less appealing physically, it is much easier to understand. User speeds are easily calculated, unlike for the waypoint model, and, as we will observe, user positions and directions are uniformly distributed. The contribution of this paper is to establish this last property for a rich class of random direction models that allow future movements to depend on past movements. To this end, we consider finite one- and two-dimensional spaces. We consider two variations, the random direction model with wrap around and with reflection. We establish a simple relationship between these two models and, for both, show that positions and directions are uniformly distributed in steady-state for a class of Markov movement models regardless of initial position. In addition, we establish a sample path property for both models, namely that any piecewise linear movement applied to a user preserves the uniform distribution of position and direction provided that users were initially uniformly throughout the space with equal likelihood of being pointed in any direction.