I will discuss a new local limit theorem on the Heisenberg group which applies to arbitrary measures of compact support and, under a mild condition on the characteristic function, obtains an optimal rate of convergence. The method extends to give mixing results for a natural class of random walks on the group $U_n(\mathbb{Z}/p\mathbb{Z})$ of $n \times n$ upper triangular matrices with entries from $\mathbb{Z}/p\mathbb{Z}$. Joint work with Persi Diaconis.