Protein aggregation is an important phenomenon that alternatively is part of the normal functioning of nature or, central to this review, has negative consequences via its hypothesized central role in neurodegenerative diseases. A key to controlling protein aggregation is understanding the mechanism(s) of protein aggregation. Kinetic studies, data curve-fitting, and analysis are, in turn, keys to rigorous mechanistic studies. The main goal of this review is to analyze and report on the primary literature contributions to protein aggregation kinetics, mechanism, and curve-fitting. Following a brief introduction, the multiple different physical methods that have been employed to follow protein aggregation are presented and briefly discussed. Next, key information on the starting proteins and especially the products, and any detectable intermediates, involved in protein aggregation are presented. This is followed by tabulation (in the Supporting information) and discussion (in the main text), of the many approaches in the literature striving to determine the kinetics and mechanism of protein aggregation. It is found that these approaches can be broadly divided into three categories: (i) kinetic and thermodynamic, (ii) empirical, and (iii) other approaches. The first two approaches are the main focus of the present contribution, their goal being curve-fitting the available kinetic data and obtaining quantitative rate constants characterizing the nucleation, growth, and any other parts of the overall aggregation process. The large literature of protein aggregation is distilled down to five classes of postulated mechanisms: i) the subsequent monomer addition mechanism, ii) the reversible association mechanism, iii) prion aggregation mechanisms, iv) an "Ockham's razor"/minimalistic model first presented in 1997 and known as the Finke-Watzky 2-step model, and v) quantitative structure activity relationship models. These five classes of mechanisms are reviewed in detail in historical order; where possible corresponding kinetic equations, and fits to aggregation data via the proposed mechanisms, are analyzed and discussed. The five classes of mechanisms are then analyzed and discussed in terms of their similarities and differences to one another. Also included is a brief discussion of selected empirical approaches used to investigate protein aggregation. Three problem areas in the protein aggregation kinetic and mechanistic studies area are identified, and a Summary and Conclusions section is provided en route to moving the field forward towards the still unachieved goal of unequivocal elucidation of the mechanism(s) of protein aggregation.