For over sixty years, researchers from across the world have been collaborating in a vast endeavour within what is known as “generative linguistics”, to describe human languages. The founding question of this research is related to the unlimited nature of linguistic possibilities. It has led to the formulation of hypotheses on linguistic combinatorics, its invariant core, and the mechanisms of variation across languages, and to the production of a precise mapping of the structures under study. Although complex, these structures can be reduced to the recursive application of extremely simple mechanisms, which are studied in the Minimalist Programme. This area of research also draws on the illuminating research on language acquisition carried out in an interdisciplinary cognitive science framework.