No detailed step-by-step model of protein rearrangements during amyloid structure formation has been presented in the literature. The aim of this work was to design a kinetic model for description of the amyloid formation process on the basis of the most recent experimental data. A general kinetic model is proposed for description of the amyloid formation process including the nucleation mechanism of polymerization with consecutive monomer attachment to oligomer and autocatalytic growth of amyloid aggregates implying all types of exponential growth such as branching, fragmentation, and growth from the surface. Computer simulations have shown that the model correctly describes experimentally observed growth stages of amyloid fibrils and that the presence of exponential growth stage in the model is critical for modeling amyloid fibril formation. The key feature of the proposed model is the stage of the exponential growth of the aggregate. Such stage can simultaneously describe several versions of aggregate enlargement by branching, fragmentation, or growth from the surface. Data obtained using this model suggest conclusions concerning the significance of each stage in amyloid fibril assembly.