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Computer simulation is used to study the competition between protein folding and aggregation, especially the formation of ordered structures that are also known as amyloid fibrils. Employing simplified protein models, we simulate multi-protein systems at a greater level of detail than has previously been possible, probe the fundamental physics that govern protein folding and aggregation, and explore the energetic and structural characteristics of amorphous and fibrillar protein aggregates. We first tackle the aggregation problem by using a low-resolution model called the lattice HP model developed by Lau and Dill. Dynamic Monte Carlo simulations are conducted on a system of simple, two-dimensional lattice protein molecules. We investigate how changing the rate of chemical or thermal renaturation affects the folding and aggregation behavior of the model protein molecule by simulating three renaturation methods: infinitely slow cooling, slow but finite cooling, and quenching. We find that the infinitely slow cooling method provides the highest refolding yields. We then study how the variation of protein concentration affects the refolding yield by simulating the pulse renaturation method, in which denatured proteins are slowly added to the refolding simulation box in a stepwise manner. We observe that the pulse renaturation method provides refolding yields that are substantially higher than those observed in the other three methods even at high packing fractions. We then investigate the folding of a polyalanine peptide with the sequence Ac-KA14K-NH2 using a novel off-lattice, intermediate-resolution protein model originally developed by Smith and Hall. The thermodynamics of a system containing a single Ac-KA14K-NH2 molecule is explored by employing the replica exchange simulation method to map out the conformational transitions as a function of temperature. We also explore the influence of solvent type on the folding process by varying the relative strength of the sidechain's hydrophobic interactions and backbone hydrogen bonding interactions. The peptide in our simulations tends to mimic a real polyalanine in that it can exist in three distinct structural states: alpha-helix, beta-structure, and random coil, depending upon the solvent conditions. We next examine the formation of fibrillar protein aggregates, which have been implicated in the pathology of several neurodegenerative diseases including Alzheimer's and Parkinson's, using the Smith/Hall intermediate-resolution protein model. Simulations were conducted on systems containing 12 to 96 Ac-KA14K-NH2 peptides at a wide variety of concentrations and temperatures. We are able to observe the formation of fibrils from random coils within just a few days on a single processor of an AMD Athlon MP 2200+ workstation. To our knowledge, these are the first simulations to span the whole process of fibril formation from the random coil state to the fibril state on such a large system. We find that fibril formation strongly depends upon the peptide concentration, the temperature, and the hydrophobic interaction strength of non-polar sidechains. The fibrils observed in our simulations mimic the structural characteristics observed in experiments in that most peptides in our fibrils were arranged in an in-register parallel orientation with intra-sheet and inter-sheet distances that are similar to those observed in experiments, and are disproportionately long along the fibril axis with about six beta-sheets, each of which contains many peptides. We also investigate the kinetics of fibril formation by performing constant-temperature simulations on systems containing 48 Ac-KA_14K-NH2 peptides with the Smith/Hall intermediate-resolution protein model. We find that fibril formation is nucleation dependent with an ordered nucleus of two beta-sheets, each with two to three peptides. The lag time before fibril formation commences decreases with increasing concentration and increases with increasing temperature. In addition, fibril formation appears to be a nucleated conformational conversion process in which small amorphous aggregates --> beta-sheets --> ordered nucleus --> subsequent rapid growth of a stable fibril. Fibril growth in our simulations involves both beta-sheet elongation, in which the fibril grows by adding individual peptides to the end of each beta-sheet and lateral addition, in which the fibril grows by adding already-formed beta-sheets to its side. Moreover, the rate of fibril formation increases with increasing concentration and decreases with increasing temperature. Finally, we examine the thermodynamics of systems containing 96 Ac-KA14K-NH2 peptides by performing replica exchange simulations over a wide range of temperatures and peptide concentrations. We map out a phase diagram in the temperature-concentration plane delineating the regions where random coils, alpha-helices, beta-sheets, fibrils, and amorphous aggregates are stable.