The Heisenberg group is the group of 3 x 3 uni-upper triangular matrices with entries in the reals,integers or mod n. This is in some sense the simplest non-commutative group. It is basic object of study in quantum physics, number theory,harmonic analysis and makes appearences in complexity theory. In joint work with Dan Bump, Angela Hicks, Laurent Miclo and Harold Widom we study simple random walk. For the integers mod n, we show that order n2 steps are necessary and sufficient for convergence. For the integers, we get sharp rates for the behavior of the chance of returning to zero. These problems have been extensively studied. One novelty, This talk uses the apparently straight forward approach of Fourier analyis. This turns out to be surprisingly difficult, needing new techniques for bounding the spectrum of symmetric tri diagonal matrices with general entries. The good news is that these same matrices come up in a host of other problems (random walks on other groups,Harpers operator,Hofstaders butterfly,the 10 martini’s problem) and our results seem new and useful.