In this thesis, optimality results are presented for Bayesian problems of sequential hypothesis testing. Conditions are given which are sufficient to demonstrate the existence and optimality of threshold policies and others are given which help characterize these policies. The general results are applied to solve four specific problems where the observations respectively arise from a time-homogeneous diffusion, a progressive semimartingale obasrved through a diffusion, a time-homogeneous Poisson process, and a predictable semimartingale observed through a point process. It is shown that threshold policies are optimal in all four cases. Exact formulae for the Bayesian costs in the point process cases will be presented for the first time.