Child language acquisition, one of Nature's most fascinating phenomena, is to a large extent still a puzzle. Experimental evidence seems to support the view that early language is highly formulaic, consisting for the most part of frozen items with limited productivity. Fairly quickly, however, children find patterns in the ambient language and generalize them to larger structures, in a process that is not yet well understood. Computational models of language acquisition can shed interesting light on this process. This paper surveys various works that address language learning from data; such works are conducted in different fields, including psycholinguistics, cognitive science and computer science, and we maintain that knowledge from all these domains must be consolidated in order for a well-informed model to emerge. We identify the commonalities and differences between the various existing approaches to language learning, and specify desiderata for future research that must be considered by any plausible solution to this puzzle.