The asymptotic mixing analysis of random walk on a group became popular in the 80's and is a growing field, with applications to cryptography, statistical physics and computer algebra systems. I will discuss several recent results resolving models which have been open since the 80's including a new local limit theorem on nilpotent groups, the asymptotic mixing of the abelian sandpile model on a square grid and the randomization of a 15-puzzle. The methods use Fourier techniques, including van der Corput's inequality and the Gowers-Cauchy-Schwarz inequality.