This thesis examines what expert and experienced PhD supervisors in the social sciences do well and how they do it. It is set in the context of the Economic and Social Research Council (ESRC) initiatives in the early 1990s to broaden the purposes of an academic research training and to promote timely PhD submissions. Many have claimed that PhD supervisors play a central role in the PhD process and this research aspired to achieve clearer understandings of the expertise involved in fulfilling that role. The research was informed by Schutz's phenomenological analysis of common sense and related concepts. It involved a student survey and six supervisor case studies. The survey aimed to determine the criteria in terms of which students judged supervision to be successful, and to identify those expert supervisors who most fully met these criteria. In going beyond criteria identified by students, the aim of the case studies was to ask how successful supervision could be achieved. 'Expert' supervisors agreeing to participate were observed over several supervision sessions and asked later in interview to talk about the various actions they took in the observed sessions. Conclusions drawn from the student survey and the case studies included a close match between student and supervisor criteria and priorities for supervision. A clear emphasis was placed by both supervisors and students on bridging gaps between student knowledge, skills and motivation at any stage and what was necessary to achieve success in their PhD studies. The distinctive nature of supervisory expertise and the willingness of supervisors to reflect usefully on their taken for-granted expert practices were thought to have important implications for the initial and continuing education of PhD supervisors, the relationships between supervision and formal research training, ESRC research training policy, and future research on the craft of PhD supervision.