With the increasing demand for doctoral education, co-supervision, understood as the formally agreed supervision of a research student by two or more academics in doctoral programmes, has become common practice in postgraduate circles in the UK. If supervision with one supervisor is complex due to personal, academic, ethical and sometimes cross-cultural issues, having two supervisors makes this process sufficiently challenging in practice to be specifically investigated in research, not least because of the additional communication issues. However, co-supervision is under-explored in the academic literature. In this article we look at the experience of co-supervision as reported by co-supervisors and those supervised by them in a UK university department within an arts and social sciences faculty, and aim to contribute to the literature on co-supervision by considering co-supervisors' and their supervisees' perspectives on co-supervision practices. Amid a general welcoming of the practice, with both parties seeing co-supervision entailing learning opportunities- for co-supervisors, learning from colleagues; for supervisees, learning from two experienced researchers--we report shared and specific concerns of these two groups. Time is a concern for both groups, but in different ways. Particularly interesting is the issue of harmony between the co-supervisors, including in feedback, the desirability of which will be perceived differently within any co-supervisor-supervisee relationship. The need for awareness-raising for co-supervisors as regards what their supervisees may feel but may not articulate may be greater for co-supervision than solo-supervision arrangements, given the additionally complex web of institutional and interpersonal relationships co-supervision entails.