We propose a kinetic model for the self-aggregation by amyloid proteins. By extending several well-known models for protein aggregation, the time evolution of aggregate concentrations containing r proteins, denoted c(r)(t), can be written in terms of generalized Smoluchowski kinetics. With this approach, we take into account all possible aggregation and fragmentation reactions involving clusters of any size. Correspondingly, an aggregate of size x + y could be formed by or break up into two smaller constituent aggregates of sizes x and y. The rates of each aggregation or fragmentation reaction, called kernels, are specified in terms of the aggregate size, and we solve c(r)(t) for large cluster sizes using numerical techniques. We show that by using Smoluchowski kinetics many pathways to fibrillation are possible and quantities, such as the aggregate length distribution at an arbitrary time, can be calculated. We show that the predicted results of the model are in agreement with the experimental observations.