In 2008 an article in this journal introduced the concept of a "fully-functioning university". This new article explores the sort of higher education (HE) that such a university would offer. It starts by examining the idea of a fully-functioning university and its relationship with the "tripartite mission" of a university. In so doing, it identifies three forms of HE that correspond to each of the three parts of the tripartite mission: subject-centred HE, student-centred HE and society-centred HE respectively. The article concludes that the sort of HE offered by a fully-functioning university would be one that comprises a subject-centred part, a student-centred part and a society-centred part. In other words, it would be a university education that prepares students to contribute to the advancement of knowledge in their subject of study (through knowledge creation, dissemination or application), to their own advancement as individuals and to the advancement of society more widely.