Despite their importance in understanding glaciological processes and constraining large-scale flow patterns in palaeo-glaciology, there is little consensus as to how drumlins are formed. Attempts to solve the ‘drumlin problem’ often fail to address how they are created from an initially flat surface in the absence of obvious cores or obstacles. This is a key strength of the instability theory, which has been described in a suite of physically-based mathematical models and proposes that the coupled flow of ice and till causes spontaneous formation of relief in the till surface. Encouragingly, model predictions of bedform height and length are consistent with observations and, furthermore, the theory has been applied to a range of subglacial bedforms and not just drumlins. However, it has yet to confront the myriad observations relating to the composition and internal structure of drumlins and this could be seen as a major deficiency. This paper is a first attempt to assess whether the instability theory is compatible with the incredible diversity of sediments and structures found within drumlins. We summarise the underlying principles of the theory and then describe and attempt to explain the main types of drumlin composition (e.g. bedrock, till, glaciofluvial sediments, and combinations thereof). Contrary to a view which suggests that the presence of some sedimentary sequences (e.g. horizontally stratified cores) is inconsistent with the theory, we suggest that one would actually expect a diverse range of constituents depending on the inheritance of sediments that pre-date drumlin formation, the duration and variability of ice flow, and the balance between erosion and deposition (till continuity) at the ice–bed interface. We conclude that the instability theory is compatible with (and potentially strengthened by) what is known about drumlin composition and, as such, offers the most complete and promising solution to the drumlin problem to date.