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Statistical learning plays a key role in language acquisition and development, from word segmentation to grammar learning. In a recent review and meta-analysis, Frost et al. (2019) identified key contributions of the statistical learning literature over the last 20 years, as well as a number of limitations. Here we address three of those limitations across three experiments. First, we address the issue of unrealistic learning environments in previous statistical learning research by training participants on an artificial language comprising multiple regularities (phonological, distributional, semantic), unlike the majority of previous statistical learning studies. Second, to examine learning at several levels of linguistic structure, we use a word learning paradigm at training, which allowed us to assess both word and grammar learning, including generalization of the trained regularities to previously unseen items. Third, to address the issue of underspecification of cognitive mechanisms underpinning statistical learning, we examine the emergence and role of explicit knowledge in generalization performance in both child and adult learners. Additionally, we examine the role of off-line memory consolidation processes. Across three experiments and multiple tasks, we found that both children and adults showed good levels of word learning, but variable levels of generalization of the trained grammatical regularities. Generalization success depended on the age group, type of training, and type of regularity assessed. Across all three experiments, explicit knowledge of the regularities contributed to the performance in some generalization tasks, but it was not key for successful generalization. Off-line consolidation processes consistently influenced long-term maintenance of the newly acquired lexical knowledge, but evidence of their role in grammar learning was mixed. We argue that our findings shed light on the cognitive mechanisms underpinning statistical learning, and provide evidence in support of multicomponential views of statistical learning.