Cognitive science has demonstrated that multilinguals (including children) show a cognitive advantage over monolinguals. Linguistics has provided evidence that multilinguals (including children) are able to successfully separate their lexicons and grammars and negotiate multilingual environments. Apart from these achievements, linguistics has generally failed to demonstrate a multilingual advantage related to the multilingual's linguistic proficiency. The present article summarizes the current literature, which shows that there are first indications of an acceleration effect in multilingual children. This effect is discernable if the languages radically differ, if the child uses a 'weak' language (often a minority language), if the child acquires more than two languages from birth, and if contact with the language exhibiting the acceleration effect is delayed until kindergarten age. This kind of acceleration effect represents an explanation gap under current theorizing in cognitive science and linguistics, and calls for a new language acquisition theory, a best-of-breed solution for further research in language acquisition. AAiMLL (Acquisition Advantages in MultiLingual Learners) combines cognitive and linguistic aspects with a threshold theory. It is claimed here that the multilingual child learns from two cognitive acquisition strategies. One is fed by grammatical features, and requires the child to (re-)use already acquired knowledge by generalizing to new domains. The other strategy enables the child to consider rejected alternatives of earlier decisions in one language, for use in the other language. An acceleration effect related to the multilingual's linguistic proficiency is indicative of the success of both strategies. The success of the strategies is argued to be related to a threshold of language usage from a quantitative or a qualitative perspective. [ABSTRACT FROM AUTHOR]