Abstract: This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.