Panel data are observations of a continuous-time process at arbitrary times, for example, visits to a hospital to diagnose disease status. Multi-state models for such data are generally based on the Markov assumption. This article reviews the range of Markov models and their extensions which can be fitted to panel-observed data, and their implementation in the msm package for R. Transition intensities may vary between individuals, or with piecewise-constant time-dependent covariates, giving an inhomogeneous Markov model. Hidden Markov models can be used for multi-state processes which are misclassified or observed only through a noisy marker. The package is intended to be straightforward to use, flexible and comprehensively documented. Worked examples are given of the use of msm to model chronic disease progression and screening. Assessment of model fit, and potential future developments of the software, are also discussed.