Protein structure is dynamic: the intrinsic flexibility of polypeptides facilitates a range of conformational fluctuations, and individual protein chains can assemble into complexes. Proteins are also dynamic in evolution: significant variations in secondary, tertiary and quaternary structure can be observed among divergent members of a protein family. Recent work has highlighted intriguing similarities between these structural and evolutionary dynamics occurring at various levels. Here we review evidence showing how evolutionary changes in protein sequence and structure are often closely related to local protein flexibility and disorder, large-scale motions and quaternary structure assembly. We suggest that these correspondences can be largely explained by neutral evolution, while deviations between structural and evolutionary dynamics can provide valuable functional insights. Finally, we address future prospects for the field and practical applications that arise from a deeper understanding of the intimate relationship between protein structure, dynamics, function and evolution.