About thirty years ago, the thorough study of migrants' initial L2 acquisitional stages in the ESF project gave birth to the notion of the Basic Variety, a simple yet autonomous language system, efficient and well suited to many communicative purposes, which learners develop in the context of untutored acquisition in immersion. Our paper discusses this notion in the light of subsequent studies which adopted a similar view on learner varieties and applied it to different populations and learning situations. Our goal is to determine whether and to what extent the core features identified for the Basic Variety need to be adapted when different variables are considered, such as the instructional context, learners' level of literacy, and the specificities of their source and target language.