I perform computational studies of quasistatically jammed packings of frictionless particles of a variety of shapes. Each of the four studies presented in this dissertation focuses primarily on the connection between the mechanical and geometric properties of these packings – specifically, how physical aspects such as force and torque balance, pressure, and shear and bulk moduli correlate with (a) constituent particle shape, (b) the degree of particle ordering in packings, and (c) the packings’ networks of interparticle contacts. I also investigate how each of these geometric properties affect each other. For example, in the first study, I examine the relationship between constituent particle shape, packing fraction, and contact number, and demonstrate a variety of correlations between these properties for two-dimensional jammed packings of nonspherical particles. In the second study, this analysis is extended to three dimensions, where an additional shape parameter is necessary to describe the packing fraction at jamming onset, and the effect of orientational ordering is considered. In the third study, I show that the scaling exponents of the ensemble-averaged shear modulus of jammed disk packings as a function of pressure are dependent on changes in the contact network that occur as the pressure increases, and demonstrate that jammed packings can unjam via isotropic compression. Finally, in the fourth study, I present the preliminary results of research into cyclically-compressed jammed disk packings, which are able to be trained into reversible states over time if the amplitude of the compression is sufficiently small, but which remain irreversible indefinitely if the amplitude of the compression is too large.