Despite a long history, there remain many important open questions about, not just the best description of structure in liquids and glasses, but what use these structures provide in terms of understanding the properties of amorphous materials. This talk will introduce the basic questions concerning the role of structure in materials science, how that structure is characterised and then present recent results on how the geometry of the locally stable structures in an amorphous materials influence the stability of the material with respect to crystallization.A central conclusion of this research is that advances in the study of amorphous structure will involve abandoning the traditional descriptive geometrical approach to structure in favour of regarding structure in terms of the statistical correlations between local structural elements. It is hoped that the description of this exciting open problem will be both accessible to and of interest for mathematicians.