Speech and language are considered uniquely human abilities: animals have communication systems, but they do not match human linguistic skills in terms of recursive structure and combinatorial power. Yet, in evolution, spoken language must have emerged from neural mechanisms at least partially available in animals. In this paper, we will demonstrate how our understanding of speech perception, one important facet of language, has profited from findings and theory in nonhuman primate studies. Chief among these are physiological and anatomical studies showing that primate auditory cortex, across species, shows patterns of hierarchical structure, topographic mapping and streams of functional processing. We will identify roles for different cortical areas in the perceptual processing of speech and review functional imaging work in humans that bears on our understanding of how the brain decodes and monitors speech. A new model connects structures in the temporal, frontal and parietal lobes linking speech perception and production.