|Abstract||Due to the apparent ease with which they can be numerically simulated, the one-dimensional self-gravitating sheet system (OGS) was first introduced by astronomers to explore different modes of gravitational evolution; including violent relaxation and the approach to thermal equilibrium. Careful work by dynamicists and statistical physicists have shown that basic claims made by astronomers regarding these models were incorrect. Unusual features of the evolution include long-lasting large-scale structures that can be thought of as one-dimensional analogues of Jupiter's red spot or a galactic spiral density wave or bar. The existence of these structures demonstrates that gravitational evolution is not entirely dominated by the second law of thermodynamics and may also contradict the Arnold diffusion ansatz. Thus the OGS can be thought of as the non-extensive analogue of the Fermi-Pasta-Ulam model of dynamical systems. This dissertation details three separate studies designed to probe the dynamical and stochastic properties of the OGS. These studies make use of stochastic modeling, local and global time averaging, and temporal and spatial correlation functions for the equal mass system, and in addition, equipartition and mass segregation for the two mass system. The results indicate that global measures of the macroscopic behavior appear to be converging to their equilibrium values in a finite time, while other measures that represent the more local properties of the system are not. It is possible, however, that these may still converge on a timescale beyond those of current simulations.