This dissertation studies the problem of assigning indivisible items to agents without monetary transfers. Pseudomarket mechanisms are proposed for both static and dynamic assignment problems. In a static setting, a new algorithm for computing pseudomarket equilibria is described. Using data from the school choice program in Seattle, the pseudomarket mechanism is compared to the Probabilistic Serial mechanism. In a dynamic setting, the pseudomarket model is extended to incorporate a continuum of agents and stochastic individual states. Conditions for equilibrium efficiency are established.