Please note special day, time and location.    We consider a nonlocal diffusion problem on a manifold. This kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. After briefly considering existence and uniqueness of solutions, we prove that, for a convenien rescaling the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior: while on compact manifolds the asymptotics are given by the spectral properties of the operator; in the model of hyperbolic space we find a different and interesting behavior. This is joint work with C. Bandle, M. Fontelos and N. Wolanski.