We motivate the use of convergence diagnostic techniques for Markov Chain Monte Carlo algorithms and review various methods proposed in the MCMC literature. A common notation is established and each method is discussed with particular emphasis on implementational issues and possible extensions. The methods are compared in terms of their interpretability and applicability and recommendations are provided for particular classes of problems. 1 Introduction There are many important implementational issues associated with MCMC methods. These include (amongst others) the choice of sampler, the number of independent replications to be run, the choice of starting values and both estimation and efficiency problems. In practice, we use ergodic averages over realisations of a Markov chain to estimate functionals of interest. In order to reduce the possibility of bias caused by the effect of starting values, iterates within an initial transient phase or burn in period are usually discarded. One o...