  Motivated by applications in Markov chain Monte Carlo, we discuss what it
means for one Markov chain to be an approximation to another. Specifically
included in that discussion are situations in which a Markov chain with
continuous state space is approximated by one with finite state space. A simple
sufficient condition for close approximation is derived, which indicates the
existence of three distinct approximation regimes. Counterexamples are
presented to show that these regimes are real and not artifacts of the proof
technique. An application to the ``ball walk'' of Lov\'asz and Simonovits is
provided as an illustrative example.
