In this article, we present a narrative review of mathematics education research on language and on communication over 2019-2022, but also look ahead by addressing challenges posed by the lack of distinction between language and communication. The persistence and significance of the problem of the distinction between language and communication are thus outlined in a historical moment of celebration of growth of research in the domain. Informed by the analysis of a selection of research journal articles and by our trajectories, we discuss influential topics in the recent discourse: multilingual mathematics classrooms; mathematics teacher education on language in mathematics teaching; multimodal mathematical communication; interaction and mathematics learning; mathematical language and discourse. We connect this with new emerging or old revisited concepts: instructional designing, gesturing, argumenting and languaging. We finish by further reflecting on multimodal mathematical communication and gesturing, and on the potential of expanding the notion of mathematics register towards a notion of mathematics communication register.