Rapid mixing

Summary form only given. In the past decade many proofs of tractability, have involved showing that some Markov chain "mixes rapidly". We do a fast tour of the highlights of Markov chain mixing, with a view toward answering, or at least addressing, the following questions: What is rapid mixing? How do you prove it, and why would you want to? Does it really have anything to do with computational complexity? And, most disturbing: is there any truth to the rumor that complexity and rapid mixing are related to phase transitions in statistical physics?.