We discuss the scope of fitting hidden Markov models (HMMs) to animal movement data in discrete time. HMMs can readily be used for fitting a variety of multi-state random walks to wildlife data. Basic discrete-time HMMs are limited to observations that are regularly spaced in time, and for which the measurement error is negligible. However, in particular for data related to terrestrial animals these conditions are often met, so that the classical likelihood-based HMM approach is feasible. We describe in detail how to apply HMMs to animal movement data, and outline a number of extensions that have not been discussed previously in the ecological literature, including incorporating individual random effects and more flexible state transition models. To demonstrate the methods we consider an application of a hierarchical hidden semi-Markov model to multiple bison movement paths.