The neuroscience of language uses experimental methodologies from cognitive science and neuroscience to investigate the neurobiological basis of linguistic phenomena in the human brain. In this chapter, we review neuroanatomical evidence for the human capacity to handle linguistic hierarchies, in line with the Chomskyan view of language as a biologically determined system computing abstract relations between words to generate grammatical linguistic sequences. We first focus on seminal neurological lesion studies assessing specific language impairments like agrammatism in patients with Broca’s aphasia. We stress the impact that this work has had on the development of neurolinguistics by highlighting the need to go beyond distinctions between language production and comprehension to investigate language competence at the basis of grammatical knowledge. In the central part of the chapter, we review current neuroscientific perspectives on the core aspects of human language put forward within the generative framework: universal principles of grammar, constituency, recursion, and Merge. We will provide evidence in favor of a fronto-temporal network in the left hemisphere comprising the connection between Brodmann area (BA) 44, the posterior portion of Broca’s area, and the posterior temporal cortex along a dorsal fiber track crucial for syntactic processing. The temporal dynamics driving the internal construction of hierarchical linguistic structure will be also introduced. An overview of maturational stages of the dorsal pathway and their relevance for the mastering of syntax will then be sketched out. We conclude by putting forward the hypothesis that the dorsal fiber tract connecting BA 44 to the posterior temporal cortex may constitute a crucial neurological precondition for the emergence of the human capacity of handling hierarchical linguistic structures. On this account, we believe that Chomsky’s notion of language as a biological system and the study of grammatical competence as distinct from performance factors have had and will continue to have profound implications for neuroscientific approaches to the study of language. Therefore, increasing collaboration between linguistics and neuroscience is strongly desirable to bring the relation between neural data and linguistic phenomena to a deeper level of understanding.