Neurocognitive and neurolinguistics theories make explicit statements relating specialized cognitive and linguistic processes to specific brain loci. These linking hypotheses are in need of neurobiological justification and explanation. Recent mathematical models of human language mechanisms constrained by fundamental neuroscience principles and established knowledge about comparative neuroanatomy offer explanations for where, when and how language is processed in the human brain. In these models, network structure and connectivity along with action- and perception-induced correlation of neuronal activity co-determine neurocognitive mechanisms. Language learning leads to the formation of action perception circuits (APCs) with specific distributions across cortical areas. Cognitive and linguistic processes such as speech production, comprehension, verbal working memory and prediction are modelled by activity dynamics in these APCs, and combinatorial and communicative-interactive knowledge is organized in the dynamics within, and connections between APCs. The network models and, in particular, the concept of distributionally-specific circuits, can account for some previously not well understood facts about the cortical ‘hubs’ for semantic processing and the motor system's role in language understanding and speech sound recognition. A review of experimental data evaluates predictions of the APC model and alternative theories, also providing detailed discussion of some seemingly contradictory findings. Throughout, recent disputes about the role of mirror neurons and grounded cognition in language and communication are assessed critically.