Thanks to Monte Carlo methods and modern computing power Bayesian inference is more accessible to practitioners than ever. The ability to sample distributions with intractable normalization constants is crucial in spatial statistics, molecular dynamics, statistical mechanics, and more. At the same time, our samplers are taken from a class of processes that are themselves interesting models; the Bayesian notion of uncertainty for hypotheses still respects the Law of Large Numbers. New sampling methods allow us to explore alternative models for more efficient inference, with one example being the advent of non-reversible Monte Carlo methods such as piecewise deterministic Markov processes (PDMPs). The purpose of this project is to develop new, accessible tools and theory for attacking difficult inference problems with and about continuous-time Markov processes.