The dynamic version of the minimum cost multicommodity flow problem that generalizes the static minimum cost multicommodity flow problem is formulated and studied. This dynamic problem is considered on directed networks with a set of commodities, time-varying capacities, fixed transit times on arcs, and a given time horizon. We assume that cost functions, defined on edges, are nonlinear and depend on time and flow and the demand function also depends on time. The corresponding algorithm, based on reducing the dynamic problem to a static problem on a time-expanded network, to solve the minimum cost dynamic multicommodity flow problem is proposed and some details concerning its complexity are discussed. Mathematics Subject Classification 2000: 90B10, 90C35, 90C27.