Amyloid fibrils are found in association with at least two dozen fatal diseases. The tendency of numerous proteins to convert into amyloid-like fibrils poses fundamental questions for structural biology and for protein science in general. Among these are the following: What is the structure of the cross-beta spine, common to amyloid-like fibrils? Is there a sequence signature for proteins that form amyloid-like fibrils? What is the nature of the structural conversion from native to amyloid states, and do fibril-forming proteins have two distinct stable states, the native state and the amyloid state? What is the basis of protein complementarity, in which a protein chain can bind to itself? We offer tentative answers here, based on our own recent structural studies.