Evolution has fine-tuned proteins to accomplish a variety of tasks. Yet, with aging, some proteins assemble into harmful amyloid aggregates associated with neurodegenerative diseases, such as Alzheimer's disease (AD), which presents a complex and costly challenge to our society. Thus, far, drug after drug has failed to slow the progression of AD, characterized by the self-assembly of the 39-43 amino acid β-amyloid (Aβ) protein into extracellular senile plaques that form a cross-β structure. While there is experimental evidence that the Aβ small oligomers are the primary toxic species, standard tools of biology have failed to provide structures of these transient, inhomogeneous assemblies. Despite extensive experimental studies, researchers have not successfully characterized the nucleus ensemble, the starting point for rapid fibril formation. Similarly scientists do not have atomic data to show how the compounds that reduce both fibril formation and toxicity in cells bind to Aβ42 oligomers. In this context, computer simulations are important tools for gaining insights into the self-assembly of amyloid peptides and the molecular mechanism of inhibitors. This Account reviews what analytical models and simulations at different time and length scales tell us about the dynamics, kinetics, and thermodynamics of amyloid fibril formation and, notably, the nucleation process. Though coarse-grained and mesoscopic protein models approximate atomistic details by averaging out unimportant degrees of freedom, they provide generic features of amyloid formation and insights into mechanistic details of the self-assembly process. The thermodynamics and kinetics vary from linear peptides adopting straight β-strands in fibrils to longer peptides adopting in parallel U shaped conformations in fibrils. In addition, these properties change with the balance between electrostatic and hydrophobic interactions and the intrinsic disorder of the system. However, simulations suggest that the critical nucleus size might be on the order of 20 chains under physiological conditions. The transition state might be characterized by a simultaneous change from mixed antiparallel/parallel β-strands with random side-chain packing to the final antiparallel or parallel states with the steric zipper packing of the side chains. Second, we review our current computer-based knowledge of the 3D structures of inhibitors with Aβ42 monomer and oligomers, a prerequisite for developing new drugs against AD. Recent extensive all-atom simulations of Aβ42 dimers with known inhibitors such as the green tea compound epigallocatechin-3-gallate and 1,4-naphthoquinon-2-yl-l-tryptophan provide a spectrum of initial Aβ42/inhibitor structures useful for screening and drug design. We conclude by discussing future directions that may offer opportunities to fully understand nucleation and further AD drug development.