Human infants develop language remarkably rapidly and without overt instruction. We argue that the distinctive ontogenesis of child language arises from the interplay of three factors: domain-specific principles of language (Universal Grammar), external experience, and properties of non-linguistic domains of cognition including general learning mechanisms and principles of efficient computation. We review developmental evidence that children make use of hierarchically composed structures (‘Merge’) from the earliest stages and at all levels of linguistic organization. At the same time, longitudinal trajectories of development show sensitivity to the quantity of specific patterns in the input, which suggests the use of probabilistic processes as well as inductive learning mechanisms that are suitable for the psychological constraints on language acquisition. By considering the place of language in human biology and evolution, we propose an approach that integrates principles from Universal Grammar and constraints from other domains of cognition. We outline some initial results of this approach as well as challenges for future research.