Rapid Mixing

In the past decade many proofs of tractability have involved showing that some Markov chain "mixes rapidly." We will 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?