We study harmonic functions for the Laplace-eltrami operator on the real hyperbolic space ${}_n$. We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball ${}_n$. We then study the Hardy spaces $H^p{(}_n)$, 0