Werner Kirsch, Peter Stollmann, G\"unter Stolz Anderson Localization for Random Schr\"odinger Operators with Long Range Interactions (42K, LaTeX) ABSTRACT. We prove pure point spectrum at all band edges for Schr\"odinger Operators with a periodic potential plus a random potential of the form $V_{\omega}(x) = \sum q_i(\omega) f(x-i)$ where $f$ is a long range interaction which decays at infinity like $|x|^{-m}$ for $m>3d$ respectively $m>2d$ depending on the regularity of $f$. We get power-decay for the eigenfunctions. The random variables $q_i$ are supposed to be independent and identically distributed. We suppose that their distribution has a bounded density of compact support.