Markov chain Monte Carlo algorithms are commonly used to approximate a variety of probability distributions, such as posterior distributions arising in Bayesian analysis. I will review the idea of coupling in the context of Markov chains, and how this idea not only leads to theoretical analyses of Markov chains but also to new Monte Carlo methods. In particular, the talk will describe how coupled Markov chains can be used to obtain 1) unbiased estimators of expectations and normalizing constants, 2) non-asymptotic convergence diagnostics for Markov chains, and 3) unbiased estimators of the asymptotic variance of an MCMC ergodic average.