We use Markov categories to generalize the basic theory of Markov chains and hidden Markov models to an abstract setting. This comprises characterizations of hidden Markov models in terms of conditional independences and algorithms for Bayesian filtering and smoothing applicable in all Markov categories with conditionals. When instantiated in appropriate Markov categories, these algorithms specialize to existing ones such as the Kalman filter, forward-backward algorithm, and the Rauch-Tung-Striebel smoother. We also prove that the sequence of outputs of our abstract Bayes filter is itself a Markov chain with a concrete formula for its transition maps. There are two main features of this categorical framework. The first is its abstract generality, as manifested in our unified account of hidden Markov models and algorithms for filtering and smoothing in discrete probability, Gaussian probability, measure-theoretic probability, possibilistic nondeterminism and others at the same time. The second feature is the intuitive visual representation of information flow in terms of string diagrams.