A striking difference between brain-inspired neuromorphic processors and current von Neumann processor architectures is the way in which memory and processing is organized. As information and communication technologies continue to address the need for increased computational power through the increase of cores within a digital processor, neuromorphic engineers and scientists can complement this need by building processor architectures where memory is distributed with the processing. In this paper, we present a survey of brain-inspired processor architectures that support models of cortical networks and deep neural networks. These architectures range from serial clocked implementations of multineuron systems to massively parallel asynchronous ones and from purely digital systems to mixed analog/digital systems which implement more biological-like models of neurons and synapses together with a suite of adaptation and learning mechanisms analogous to the ones found in biological nervous systems. We describe the advantages of the different approaches being pursued and present the challenges that need to be addressed for building artificial neural processing systems that can display the richness of behaviors seen in biological systems.