In Markov chain Monte Carlo, a Markov chain is constructed whose limiting distribution is equal to some target distribution. While it is easy to build such chains, for some distributions the standard constructions can take exponentially long to come near that limit, making the chain torpidly mixing. When the limit is reached in polynomial time, the chain is rapidly mixing. Tempering is a technique designed to speed up the convergence of Markov chains by adding an extra temperature parameter that acts to smooth out the target distribution. In this talk I will present joint work with Dawn Woodard (Cornell) and Scott Schmidler (Duke) that give sufficient conditions for a tempering chain to be torpidly mixing, and a related (but different) set of conditions for the chain to be rapidly mixing.