Protein molecules have the ability to form a rich variety of natural and artificial structures and materials. We show that amyloid fibrils, ordered supramolecular nanostructures that are self- assembled from a wide range of polypeptide molecules, have rigidities varying over four orders of magnitude, and constitute a class of high- performance biomaterials. We elucidate the molecular origin of fibril material properties and show that the major contribution to their rigidity stems from a generic interbackbone hydrogen- bonding network that is modulated by variable side- chain interactions.