In this talk, we'll describe Helgason's conjecture for Riemannian symmetric space of non-compact type. This is a generalisation of the classical correspondence in complex anaysis between harmonic functions on the unit disk, endowed with the hyperbolic metric, and functions on the boundary of the disk given by the Poisson integral. This was proved in full generality using sophisticated mircolocal techniques by Kashiwara, Kowata, Minemura, Okamoto, Oshima and Tanaka. We'll outline a simple proof of this result for Riemannian symmetric spaces of rank one. Time-permitting, we'll mention a nice application to scattering theory.