  The notion of $L^p$-distributions is introduced on Riemannian symmetric
spaces of noncompact type and their main properties are established. We use a
geometric description for the topology of the space of test functions in terms
of the Laplace-Beltrami operator. The techniques are based on a-priori
estimates for elliptic operators. We show that structure theorems, similar to
$\Rn$, hold on symmetric spaces. We give estimates for the convolutions.
