The aim of this talk is not mainly to discuss the physical background of these objects but instead to consider random Schroedinger operators and the IDS on more general spaces than the Euclidean space. The Schroedinger operators under consideration are defined on the universal covering of a compact Riemannian manifold with amenable group of deck transformations. The main result of the talk is a proof for the existence of a non-random IDS for random families of Schroedinger operators. It uses geometric and measure theoretic methods. Moreover, almost all operators of such a random family have the same spectrum, and this almost sure spectrum of the family coincides with the the points of increase of the IDS.