Amyloid or amyloid-like fibrils represent a general class of nanomaterials that can be formed from many different peptides and proteins. Although these structures have an important role in neurodegenerative disorders, amyloid materials have also been exploited for functional purposes by organisms ranging from bacteria to mammals. Here we review the functional and pathological roles of amyloid materials and discuss how they can be linked back to their nanoscale origins in the structure and nanomechanics of these materials. We focus on insights both from experiments and simulations, and discuss how comparisons between functional protein filaments and structures that are assembled abnormally can shed light on the fundamental material selection criteria that lead to evolutionary bias in multiscale material design in nature.