This review provides a summary of the state-of-the-art in terms of description, interpretation and classification of crystal structures in mineralogy. Among the various methods, the focus is on atomic packing (including both anion and cation arrays), coordination polyhedra (both cation- and anion-centred), the concept of fundamental building blocks and related ideas and the use of networks, graphs and tilings (space partitions). The basic concepts under discussion in modern structural mineralogy include structure hierarchy (specification and compositional hierarchies are considered separately), modularity (representation of crystal structures as constructed from modules extracted from simple archetype structures) and complexity (with emphasis on static or informational and algorithmic complexities). Short historical notes are given for all of the topics considered.