Markov processes are used in various domains to model the uncertain future behaviour of the phenomenon under study; applications range from the capital of an insurer to the number of customers in a queue. However, it is not always feasible nor desirable to exactly specify the parameters that characterise such a Markov process, and the strong simplifying assumptions underlying the model are not always justified. Imprecise (or nonlinear) Markov processes deal with these two issues by allowing partially specified parameters and relaxing some of the assumptions. While the usefulness of this generalisation is evident, the state of the art has a number of technical limitations that hinder its application in practice, and this project strives to eliminate these limitations. First, it aims to obtain a method to construct an imprecise Markov process with regular paths, so that non-finitary inferences can be dealt with. Second, it aims to come up with numerical methods based on lumping to compute (tight lower and upper bounds on) probabilities and expectations for imprecise Markov processes with a countably infinite state space. Finally, the project aims to show that specific non-finitary inferences—like expected time averages—for countable-state imprecise Markov processes are the (viscosity) solution to an ordinary differential equation, and similarly for the ruin probability in the imprecise Cramér-Lundberg process.