Abstract Neuroscience has greatly improved our understanding of the brain basis of abstract lexical and semantic processes. The neuronal devices underlying words and concepts are distributed neuronal assemblies reaching into sensory and motor systems of the cortex and, at the cognitive level, information binding in such widely dispersed circuits is mirrored by the sensorimotor grounding of form and meaning of symbols. Recent years have seen the emergence of evidence for similar brain embodiment of syntax. Neurophysiological studies have accumulated support for the linguistic notion of abstract combinatorial rules manifest as functionally discrete neuronal assemblies. Concepts immanent to the theory of abstract automata could be grounded in observations from modern neuroscience, so that it became possible to model abstract pushdown storage – which is critical for building linguistic tree structure representations – as ordered dynamics of memory circuits in the brain. At the same time, neurocomputational research showed how sequence detectors already known from animal brains can be neuronally linked so that they merge into larger functionally discrete units, thereby underpinning abstract rule representations that syntactically bind lexicosemantic classes of morphemes and words into larger meaningful constituents. Specific predictions of brain-based grammar models could be confirmed by neurophysiological and brain imaging experiments using MEG, EEG and fMRI. Neuroscience and neurocomputational research offering perspectives on understanding abstract linguistic mechanisms in terms of neuronal circuits and their interactions therefore point programmatic new ways to future theory-guided experimental investigation of the brain basis of grammar.