In the theory of wavelets, if one goes beyond the Euclidean space $\bold R^n$, the first, or perhaps the most important case is the Heisenberg group $\bold H^n$. In this talk we will discuss: (1) The admissible (continuous) wavelets on the Heisenberg group. The irreducible decompositions of $L^2$ function spaces and the reproducing kernels (H. Liu and L. Peng). (2) The orthogonal wavelets on the Heisenberg group. In fact we discover a method to construct the discrete wavelets on Heisenberg group from one dimensional wavelets which is also valid for biorthogonal wavelets and multiwavelets (preprint by B. Jawerth and L. Peng).