We derive the sharp constants for the inequalities on the Heisenberg group whose analogues on Euclidean space are the well known Hardy-Littlewood-Sobolev inequalities. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago, which was crucial in the solution of the CR Yamabe problem. Our methodology is completely different from that used to obtain the Euclidean inequalities and can be used to give a new, rearrangement free, proof of the HLS inequalities. The talk is based on joint work with E. H. Lieb.