This one-day tutorial will present the mathematical theory of Markov chain convergence, including such concepts as random walks, recurrence and transience, stationary distributions, reversibility, etc.  It will describe Brownian motion and diffusions as continuous limits of discrete Markov chains.  It will then apply this knowledge to Markov chain Monte Carlo (MCMC) algorithms, explaining their convergence and efficiency from a theoretical viewpoint.  It will explain how MCMC algorithms can converge to diffusions, and how to use that fact to optimise their performance. Depending on time, it may also discuss how to optimise tempering algorithms, and/or prove convergence of adaptive MCMC algorithms.  No background is assumed beyond basic undergraduate-level probability theory and mathematical reasoning, and perhaps a bit of familiarity with MCMC.