The self-assembly of proteins into filamentous structures underpins many aspects of biology, from dynamic cell scaffolding proteins such as actin, to the amyloid plaques responsible for a number of degenerative diseases. Typically, these self-assembly processes have been treated as nucleated, reversible polymerisation reactions, where dynamic fluctuations in a population of monomers eventually overcome an energy barrier, forming a stable aggregate that can then grow and shrink by the addition and loss of more protein from its ends. The nucleated, reversible polymerisation framework is very successful in describing a variety of protein systems such as the cell scaffolds actin and tubulin, and the aggregation of haemoglobin. Historically, amyloid fibrils were also thought to be described by this model, but measurements of their aggregation kinetics failed to match the model's predictions. Instead, recent work indicates that autocatalytic polymerisation - a process by which the number of growth competent species is increased through secondary nucleation, in proportion to the amount already present - is better at describing their formation. In this thesis, I will extend the predictions made in this mean-field, autocatalytic polymerisation model through use of kinetic Monte Carlo simulations. The ubiquitous sigmoid-like growth curve of amyloid fibril formation often possesses a notable quiescent lag phase which has been variously attributed to primary and secondary nucleation processes. Substantial variability in the length of this lag phase is often seen in replicate experimental growth curves, and naively may be attributed to fluctuations in one or both of these nucleation processes. By comparing analytic waiting-time distributions, to those produced by kinetic Monte Carlo simulation of the processes thought to be involved, I will demonstrate that this cannot be the case in sample volumes comparable with typical laboratory experiments. Experimentally, the length of the lag phase, or "lag time", is often found to scale with the total protein concentration, according to a power law with exponent γ. The models of nucleated polymerisation and autocatalytic polymerisation predict different values for this scaling exponent, and these are sometimes used to identify which of the models best describes a given protein system. I show that this approach is likely to result in a misidentification of the dominant mechanisms under conditions where the lag phase is dominated by a different process to the rest of the growth curve. Furthermore, I demonstrate that a change of the dominant mechanism associated with total protein concentration will produce "kinks" in the scaling of lag time with total protein concentration, and that these may be used to greater effect in identifying the dominant mechanisms from experimental kinetic data. Experimental data for bovine insulin aggregation, which is well described by the autocatalytic polymerisation model for low total protein concentrations, displays an intriguing departure from the predicted behaviour at higher protein concentrations. Additionally, the protein concentration at which the transition occurs, appears to be affected by the presence of salt. Coincident with this, an apparent change in the fibril structure indicates that different aggregation mechanisms may operate at different total protein concentrations. I demonstrate that a transition whereby the self-assembly mechanisms change once a critical concentration of fibrils or fibrillar protein is reached, can explain the observed behaviour and that this predicts a substantially higher abundance of shorter laments - which are thought to be pathogenic - at lower total protein concentrations than if self-assembly were consistently autocatalytic at all protein concentration. Amyloid-like loops have been observed in electron and atomic-force microscographs, together with non-looped fibrils, for a number of different proteins including ovalbumin. This implies that fibrils formed of these proteins are able to grow by fibrillar end-joining, and not only monomer addition as is more commonly assumed. I develop a simple analytic expression for polymerisation by monomer addition and fibrillar end-joining, (without autocatalysis) and show that this is not sufficient to explain the growth curves obtained experimentally for ovalbumin. I then demonstrate that the same data can be explained by combining fibrillar end-joining and fragmentation. Through the use of an analytic expression, I estimate the kinetic rates from the experimental growth curves and, via simulation, investigate the distribution of lament and loop lengths. Together, my findings demonstrate the relative importance of different molecular mechanisms in amyloid fibril formation, how these might be affected by various environmental parameters, and characteristic behaviour by which their involvement might be detected experimentally.