A stochastic processing network is a system that takes materials of various kinds as inputs, and uses processing resources to produce other materials as outputs. Such a network provides a powerful abstraction of a wide range of real world, complex systems, including semiconductor wafer fabrication facilities, networks of data switches, and large-scale call centers. Key performance measures of a stochastic processing network include throughput, cycle time, and holding cost. The network performance can dramatically be affected by the choice of operational policies. We propose a family of operational policies called maximum pressure policies. The maximum pressure policies are attractive in that their implementation uses minimal state information of the network. The deployment of a resource (server) is decided based on the queue lengths in its serviceable buffers and the queue lengths in their immediate downstream buffers. In particular, the decision does not use arrival rate information that is often difficult or impossible to estimate reliably. We prove that a maximum pressure policy can maximize throughput for a general class of stochastic processing networks. We also establish an asymptotic optimality of maximum pressure policies for stochastic processing networks with a unique bottleneck. The optimality is in terms of minimizing workload process. A key step in the proof of the asymptotic optimality is to show that the network processes under maximum pressure policies exhibit a state space collapse.