  This paper surveys various results about Markov chains on general
(non-countable) state spaces. It begins with an introduction to Markov chain
Monte Carlo (MCMC) algorithms, which provide the motivation and context for the
theory which follows. Then, sufficient conditions for geometric and uniform
ergodicity are presented, along with quantitative bounds on the rate of
convergence to stationarity. Many of these results are proved using direct
coupling constructions based on minorisation and drift conditions. Necessary
and sufficient conditions for Central Limit Theorems (CLTs) are also presented,
in some cases proved via the Poisson Equation or direct regeneration
constructions. Finally, optimal scaling and weak convergence results for
Metropolis-Hastings algorithms are discussed. None of the results presented is
new, though many of the proofs are. We also describe some Open Problems.
