One of the most striking features of human languages is their extreme variety. Even more striking is the existence, behind the apparent variety, of strong representational and cognitive regularities that govern their form and their function: language universals. We discuss here some recent work from our group, where large-scale, data-intensive computational modelling techniques are used to address fundamental linguistic questions on language regularities. In the area of word order, we report here on work that leverages large amounts of monolingual and parallel corpus data to develop computational models of the internal structure of the noun phrase (Universal 20) and of general structural minimisation principles. In the area of event duration, we report on work that leverages both deep similarities and surface differences to develop truly cross-linguistic natural language processing tools.