The issue of universities' reputations has been transformed by the development of international university rankings as well as by national publicly funded teaching-quality and research assessment exercises, and by independent discipline-based and institutional reviews and guides. Strictly speaking the attribution of reputational standing is not absolute but aspectual and there area number of different features in respect of a university may be well regarded. In guides and rankings, there is also an element of reputational a priorism based on shared assumptions about the nature of universities. It is worth thinking about these assumptions because they relate to what, I suggest, remains the primary role of universities, namely the teaching of undergraduates in traditional disciplines.