1. Introduction 22. Sources of structural information in solid-state NMR data 52.1 General remarks 52.2 Chemical shifts, linewidths, and magic-angle spinning 62.3 Dipole–dipole couplings and dipolar recoupling 82.4 Tensor correlation techniques 122.5 Solid-state NMR of aligned samples 142.6 Indirect sources of structural information 152.7 Sample preparation for solid-state NMR 153. Levels of structure in amyloid fibrils 184. Molecular structure of β-amyloid fibrils 254.1 Self-propagating, molecular-level polymorphism in Aβ1–40 fibrils 254.2 Structural model for Aβ1-40 fibrils 284.3 Staggering of β-strands in Aβ1-40 fibrils 324.4 Structure of Aβ1-42 fibrils 344.5 Structure of fibrils formed by short β-amyloid fragments 344.6 Structures of non-fibrillar aggregates 355. Molecular structure of other amyloid fibrils 365.1 Ure2p10–39 and full-length Ure2p fibrils 365.2 TTR105–115 fibrils 385.3 HET-s fibrils 385.4 Amylin fibrils 395.5 PrP fibrils 395.6 ccβ fibrils 405.7 α-synuclein fibrils 405.8 Calcitonin fibrils 416. Data relevant to various proposals regarding amyloid structure 416.1 β-helical models for amyloid fibrils 416.2 Amyloid fibrils as water-filled nanotubes 426.3 Domain swapping in amyloid fibrils 426.4 The parallel superpleated β-structure model 436.5 α-sheet structures in amyloid fibrils 437. Conclusions 448. Acknowledgments 469. References 46Solid-state nuclear magnetic resonance (NMR) measurements have made major contributions to our understanding of the molecular structures of amyloid fibrils, including fibrils formed by the β-amyloid peptide associated with Alzheimer's disease, by proteins associated with fungal prions, and by a variety of other polypeptides. Because solid-state NMR techniques can be used to determine interatomic distances (both intramolecular and intermolecular), place constraints on backbone and side-chain torsion angles, and identify tertiary and quaternary contacts, full molecular models for amyloid fibrils can be developed from solid-state NMR data, especially when supplemented by lower-resolution structural constraints from electron microscopy and other sources. In addition, solid-state NMR data can be used as experimental tests of various proposals and hypotheses regarding the mechanisms of amyloid formation, the nature of intermediate structures, and the common structural features within amyloid fibrils. This review introduces the basic experimental and conceptual principles behind solid-state NMR methods that are applicable to amyloid fibrils, reviews the information about amyloid structures that has been obtained to date with these methods, and discusses how solid-state NMR data provide insights into the molecular interactions that stabilize amyloid structures, the generic propensity of polypeptide chains to form amyloid fibrils, and a number of related issues that are of current interest in the amyloid field.