  Markov chain Monte Carlo (MCMC) is one of the most useful approaches to
scientific computing because of its flexible construction, ease of use and
generality. Indeed, MCMC is indispensable for performing Bayesian analysis. Two
critical questions that MCMC practitioners need to address are where to start
and when to stop the simulation. Although a great amount of research has gone
into establishing convergence criteria and stopping rules with sound
theoretical foundation, in practice, MCMC users often decide convergence by
applying empirical diagnostic tools. This review article discusses the most
widely used MCMC convergence diagnostic tools. Some recently proposed stopping
rules with firm theoretical footing are also presented. The convergence
diagnostics and stopping rules are illustrated using three detailed examples.
