In this chapter we develop a generalization of hidden Markov models valid for the evolution of a system in continuous time. That is, we describe how to model and analyze hidden Markov jump processes. In this context, we introduce the concept of uniformization to simulate continuous time trajectories and then use uniformization to develop a Monte Carlo strategy to sample trajectories from a posterior over trajectories. Having discussed how trajectories can be sampled from the posterior over all candidate trajectories, we then describe strategies for full posterior inference over trajectories and other model parameters. We end with strategies for trajectory marginalization and continuous time filtering.