University mathematics education (UME) is considered, in this paper, as a kind of "didactic practice"--characterised by institutional settings and by the purpose of inducting students into "mathematical practices." We present a research programme -- the anthropological theory of the didactic (ATD)--in which this rough definition can be made much more precise; we also outline some cases of ATD-based research on UME. Three cases are presented in more detail. The first is a theoretical and empirical study of the topic of "dual vector spaces," as it appears in undergraduate courses on linear algebra. The second case concerns a similar study of the practices and theories on "limits of functions" which students may develop in calculus courses. Finally, a third study illustrates the use of ATD to design and experiment innovative approaches to mathematical modelling in the setting of a first mathematics course for engineering students.