Peptides and proteins have been found to possess an inherent tendency to convert from their native functional states into intractable amyloid aggregates. This phenomenon is associated with a range of increasingly common human disorders, including Alzheimer and Parkinson diseases, type II diabetes, and a number of systemic amyloidoses. In this review, we describe this field of science with particular reference to the advances that have been made over the last decade in our understanding of its fundamental nature and consequences. We list the proteins that are known to be deposited as amyloid or other types of aggregates in human tissues and the disorders with which they are associated, as well as the proteins that exploit the amyloid motif to play specific functional roles in humans. In addition, we summarize the genetic factors that have provided insight into the mechanisms of disease onset. We describe recent advances in our knowledge of the structures of amyloid fibrils and their oligomeric precursors and of the mechanisms by which they are formed and proliferate to generate cellular dysfunction. We show evidence that a complex proteostasis network actively combats protein aggregation and that such an efficient system can fail in some circumstances and give rise to disease. Finally, we anticipate the development of novel therapeutic strategies with which to prevent or treat these highly debilitating and currently incurable conditions.