Tumour formation involves random mutagenic events and positive evolutionary selection acting on a subset of such events, referred to as driver mutations. A decade of careful surveying of tumour DNA using exome-based analyses has revealed a multitude of protein-coding somatic driver mutations, some of which are clinically actionable. Today, a transition towards whole-genome analysis is well under way, technically enabling the discovery of potential driver mutations occurring outside protein-coding sequences. Mutations are abundant in this vast non-coding space, which is more than 50 times larger than the coding exome, but reliable identification of selection signals in non-coding DNA remains a challenge. In this Review, we discuss recent findings in the field, where the emerging landscape is one in which non-coding driver mutations appear to be relatively infrequent. Nevertheless, we highlight several notable discoveries. We consider possible reasons for the relative absence of non-coding driver events, as well as the difficulties associated with detecting signals of positive selection in non-coding DNA.