This paper provides a comprehensive survey of Markov chain theory on general (non-countable) state spaces, focusing on Markov chain Monte Carlo (MCMC) algorithms. It begins with an introduction to MCMC, highlighting its importance in statistics for sampling from complex probability distributions. The paper then discusses sufficient conditions for geometric and uniform ergodicity, along with quantitative bounds on the rate of convergence to stationarity. These results are often proved using direct coupling constructions based on minorization and drift conditions. The paper also covers necessary and sufficient conditions for Central Limit Theorems (CLTs), optimal scaling and weak convergence results for Metropolis-Hastings algorithms. While many of the results are not new, the proofs are often novel. The authors also describe some open problems in the field.This paper provides a comprehensive survey of Markov chain theory on general (non-countable) state spaces, focusing on Markov chain Monte Carlo (MCMC) algorithms. It begins with an introduction to MCMC, highlighting its importance in statistics for sampling from complex probability distributions. The paper then discusses sufficient conditions for geometric and uniform ergodicity, along with quantitative bounds on the rate of convergence to stationarity. These results are often proved using direct coupling constructions based on minorization and drift conditions. The paper also covers necessary and sufficient conditions for Central Limit Theorems (CLTs), optimal scaling and weak convergence results for Metropolis-Hastings algorithms. While many of the results are not new, the proofs are often novel. The authors also describe some open problems in the field.