Mobility-on-demand (MoD) services are provided by multiple competing companies. In their competition for travelers, they need to provide minimum travel costs, or travelers will switch to competitors. This study developed a dynamic traffic assignment of MoD systems. A static traffic assignment (STA) model is first defined. When demand is asymmetric, empty rebalancing trips are required to move vehicles to traveler origins, and the optimal rebalancing flows are found by a linear program. Because of the time-dependent nature of traveler demand, the model was converted to dynamic traffic assignment (DTA). The method of successive averages, which is provably convergent for STA, was used to find dynamic user equilibrium (DUE). The simulation was conducted on two networks. The MoD system was simulated with different fleet sizes and demands. The results showed that the average total delay and travel distance decreased with the increase in fleet size whereas the average on-road travel time increased with the fleet size. The result of traffic assignment of one network with MoD system was compared with a network where all travelers use private vehicles. The results showed that the network with MoD system created more trips but less traffic congestion.