Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and large scale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of such a system. LP-based planning is critical in setting a medium range or long term goal for many systems. But it does not translate into a day-to-day operational policy that must deal with discreteness of jobs and the randomness of the processing environment. A processing network, advanced by J. Michael Harrison, is a system that takes inputs of materials of various kinds and uses various processing resources to produce outputs of materials of various kinds. Such a network provides a powerful abstraction of a wide range of real world systems. It contains features rich enough to capture many important elements encountered in practice. It can model machine-operator interaction, material handling equipment, machine breakdown, and fork-and-join operations, all in a manufacturing system. It can model call centers with cross trained operators, input-queued data switches, and congestion based road traffic routing. In this talk, we will present a family of dynamic, operational policies, called maximum pressure policies, that can achieve maximum throughput predicted by LPs for a wide class of stochastic processing networks.