Human language allows us to create an infinitude of ideas from a finite set of basic building blocks. What is the neurobiology of this combinatory system? Research has begun to dissect the neural basis of natural language syntax and semantics by analyzing the basics of meaning composition, such as two-word phrases. This work has revealed a system of composition that involves rapidly peaking activity in the left anterior temporal lobe and later engagement of the medial prefrontal cortex. Both brain regions show evidence of shared processing between comprehension and production, as well as between spoken and signed language. Both appear to compute meaning, not syntactic structure. This Review discusses how language builds meaning and lays out directions for future neurobiological research on the combinatory system.