The human capacity to acquire language is an outstanding scientific challenge to understand. Somehow our language capacities arise from the way the human brain processes, develops and learns in interaction with its environment. To set the stage, we begin with a summary of what is known about the neural organization of language and what our artificial grammar learning (AGL) studies have revealed. We then review the Chomsky hierarchy in the context of the theory of computation and formal learning theory. Finally, we outline a neurobiological model of language acquisition and processing based on an adaptive, recurrent, spiking network architecture. This architecture implements an asynchronous, event-driven, parallel system for recursive processing. We conclude that the brain represents grammars (or more precisely, the parser/generator) in its connectivity, and its ability for syntax is based on neurobiological infrastructure for structured sequence processing. The acquisition of this ability is accounted for in an adaptive dynamical systems framework. Artificial language learning (ALL) paradigms might be used to study the acquisition process within such a framework, as well as the processing properties of the underlying neurobiological infrastructure. However, it is necessary to combine and constrain the interpretation of ALL results by theoretical models and empirical studies on natural language processing. Given that the faculty of language is captured by classical computational models to a significant extent, and that these can be embedded in dynamic network architectures, there is hope that significant progress can be made in understanding the neurobiology of the language faculty.