In this paper, we review the Doi-Onsager model of orientation dynamics for rod-like polymers in solution, and present additional analysis. We discuss existing results on existence and convergence of solutions, as well as properties of equilibria. In particular, this model captures a phase transition indicating a switch between aligned and disordered particle orientations. We use nonlinear and linear stability analysis to further characterize this bifurcation. A new proof of equilibrium distribution symmetry is given for the two-dimensional problem. Finally, we derive, analyze, and demonstrate a numerical scheme for the two-dimensional problem which preserves the mass, positivity, and entropy-decreasing nature of the system.