Discrete-time real-valued Markov chains are sometimes observed at certain time points only. The unobserved time points may be deterministic and periodic, or the gap length may be random and may also depend on the last observed state. This is no problem for estimating the one-dimensional marginal distribution. However, if we want to estimate joint or conditional distributions of the chain, in particular the transition distribution, can we exploit the information in pairs of observations separated by an unobserved gap? We discuss when and how this is possible in nonparametric and in autoregressive models. We point out similarities to mixture models and to regression models with observations missing at random.