The deposition of amyloid- and amyloid-like fibrils is the main pathological hallmark of numerous protein misfolding diseases including Alzheimer's disease, transmissible spongiform encephalopathy, and type 2 diabetes. Besides the well-established role in disease, recent work on a variety of organisms ranging from bacteria to humans suggests that amyloid fibrils can also convey biological functions. To better understand the molecular mechanisms by which amyloidogenic proteins misfold in disease or perform biological functions, structural information is essential. Although high-resolution structural analysis of amyloid fibrils has been challenging, a combination of biophysical approaches is beginning to unravel the various structural features of amyloid fibrils. Here we review these recent developments with particular emphasis on amyloid fibrils that have been studied using site-directed spin labeling and electron paramagnetic resonance spectroscopy. This approach has been used to define the precise location of fibril-forming core regions and identify local secondary structures within such core regions. Perhaps one of the most remarkable findings arrived at by site-directed spin labeling was that most fibrils that contain an extensive core region of 20 amino acids or more share a common parallel in-register arrangement of beta strands. The preference for this arrangement can be explained on topological grounds and may be rationalized by the maximization of hydrophobic contact surface.