Our aim is to identify the challenges of teaching music composition to today's students of composition at a tertiary or professional level. We undertake this by two different approaches: on one level, by establishing a context through reviewing literature on the teaching practices of three renowned teachers from the twentieth century; on a second level, by considering current practice in the UK through a qualitative case study of six distinguished composers involved in the professional and academic world: George Benjamin, Robert Saxton, Michael Finnissy, Judith Weir, Michael Zev Gordon and Nicola LeFanu. Together these two complementary approaches will provide our first look at an insufficiently researched area.