A common practice in the cognitive neurosciences is to investigate population-typical phenomena, treating individuals as equal except for a few outliers that are usually discarded from analyses or that disappear on group-level patterns. Only a few studies to date have captured the heterogeneity of language processing across individuals as so-called “individual differences”; fewer have explicitly researched language aptitude, which designates an individual’s ability for acquiring foreign languages. Existing studies show that, relative to average learners, very gifted language learners display different task-related patterns of functional activation and connectivity during linguistic tasks, and structural differences in white and grey matter morphology, and in white matter connectivity. Despite growing interest in language aptitude, there is no recent comprehensive review, nor a theoretical model to date that includes the neural level. To fill this gap, we review neuroscientific research on individual differences in language learning and language aptitude and present a first, preliminary neurocognitive model of language aptitude. We suggest that language aptitude could arise from an advantageous neurocognitive profile, which leads to high intrinsic motivation and proactive engagement in language learning activities. On the neural level, interindividual differences in the morphology of the bilateral auditory cortex constrain individual neural plasticity, as is evident in the speed and efficiency of language learning. We suggest that language learning success is further dependent upon highly efficient auditory-motor connections (speech-motor networks) and the structural characteristics of dorsal and ventral fibre tracts during language learning.