Random band matrices provide a simple model for studying the metal-insulator transition of the Anderson model. After giving some background on this problem I will present a recent simple proof that N×N Gaussian random band matrices with band width W exhibit dynamical Anderson localization at all energies when W≪N^{1/4}. The proof uses the fractional moment method and an adaptive Mermin-Wagner style shift. Joint with Cipolloni, Peled, and Schenker.