Protein aggregation correlates with the development of several deleterious human disorders such as Alzheimer's disease, Parkinson's disease, prion-associated transmissible spongiform encephalopathies and type II diabetes. The polypeptides involved in these disorders may be globular proteins with a defined 3D-structure or natively unfolded proteins in their soluble conformations. In either case, proteins associated with these pathogeneses all aggregate into amyloid fibrils sharing a common structure, in which beta-strands of polypeptide chains are perpendicular to the fibril axis. Because of the prominence of amyloid deposits in many of these diseases, much effort has gone into elucidating the structural basis of protein aggregation. A number of recent experimental and theoretical studies have significantly increased our understanding of the process. On the one hand, solid-state NMR, X-ray crystallography and single molecule methods have provided us with the first high-resolution 3D structures of amyloids, showing that they exhibit conformational plasticity and are able to adopt different stable tertiary folds. On the other hand, several computational approaches have identified regions prone to aggregation in disease-linked polypeptides, predicted the differential aggregation propensities of their genetic variants and simulated the early, crucial steps in protein self-assembly. This review summarizes these findings and their therapeutic relevance, as by uncovering specific structural or sequential targets they may provide us with a means to tackle the debilitating diseases linked to protein aggregation.