i will describe the three phases for the mixing time (the time to equilibriate) of the glauber dynamics for the ising model on the complete graph on $n$ vertices. at high temperature, the time required to mix is order $n(\log n)$, and there is a cut-off, meaning that in a window of order $n$, the distance to equilibrium drops from near one to near zero. at critical temperature, the dynamics mix in order $n^(3/2)$ steps. at low temperature, the mixing is exponentially slow, but if the dynamics are restricted to one of the two modes of the stationary distribution, then it mixes in order $n(\log n)$ steps. joint work with m. luczak and y. peres.