In this disseration, the author studies sequential Bayesian learning problems modeled under non-Gaussian distributions. We focus on a class of problems called the multi-armed bandit problem, and studies its optimal learning strategy, the Gittins index policy. The Gittins index is computationally intractable and approxi- mation methods have been developed for Gaussian reward problems. We construct a novel theoretical and computational framework for the Gittins index under non- Gaussian rewards. By interpolating the rewards using continuous-time conditional Levy processes, we recast the optimal stopping problems that characterize Gittins indices into free-boundary partial integro-differential equations (PIDEs). We also provide additional structural properties and numerical illustrations on how our ap- proach can be used to approximate the Gittins index.