Abstract: This dissertation describes an investigation of jammed packings of frictionless hard particles, including the computer generation of (nearly) jammed packings, the development of mathematical criteria and algorithms to verify jamming, and computational and experimental studies of disordered and ordered hard-sphere and hard-ellipsoid packings. In the first part of this dissertation a mathematical framework for understanding jamming in packings of hard particles is developed. Algorithms to model hard-particle systems, and in particular, a collision-driven molecular dynamics algorithm for the simulation of dense packings of hard spheres, ellipsoids and superellipsoids, are designed. This algorithm is used to implement a generalization of the Lubachevsky-Stillinger algorithm to generate disordered packings of hard spheres and hard ellipsoids. It is found that the density and average contact number of the random packings rises sharply, but continuously, as asphericity is introduced, leading to hypostatic packings much denser than well-known random sphere packings. A mathematical theory of jamming for packings of spherical and nonspherical particles, as well as algorithms to test whether a packing is (nearly) jammed are developed, verifying that our packings are jammed. A molecular-dynamics algorithm to calculate the (non-equilibrium) free energy of nearly jammed packings of hard particles is designed and implemented. In the second part of this dissertation the properties of disordered and ordered packings of hard particles are studied. Investigated are correlations, including short-ranged order in the pair-correlation function, as well as long-ranged density fluctuations in the structure factor, for hard sphere packings in both three and higher dimensions. An unusual multitude of near contacts persistent with dimensionality, as well as a decorrelation for distances beyond contact as dimension increases, are found. Comparisons find good agreement between computational and experimental results for packing of hard ellipsoids in finite containers. The densest known ordered packing of hard ellipsoids is discovered using molecular dynamics. Finally, the thermodynamics of dense systems of hard-particles is investigated. The phase-diagram of hard rectangles of aspect ratio two (dominos) is determined. Finally, it is demonstrated that there is no ideal glass transition for binary systems of hard disks.