What is, and what is not, a Markov chain?

This is a purely expository paper, written in response to the increasing use of Markovian models to represent hydrological phenomena. After outlining the more familiar properties of simple (1-step) Markov chains the paper describes non-simple ( k-step) chains and bivariate chains, and discusses some fallacies associated with the transition probabilities of non-Markov chains. Some attention is devoted to the problems of selecting a model for use in practical situations.