Abstract: Bombarding a solid surface with a broad ion beam can produce a remarkable variety of self-assembled nanoscale patterns, including ripples and arrays of nanodots. The anisotropic Kuramoto-Sivashinsky equation is widely used to model the formation of the ripples produced by ion bombardment. The primary obstacle that has prevented the adoption of ion bombardment as a nano-fabrication tool is the high density of defects in the patterns that typically form. Our simulations indicate that the problem could be remedied simply by rocking the sample during ion bombardment. From a mathematical standpoint, we find that a temporally periodic coefficient in the anisotropic Kuramoto-Sivashinsky equation can suppress the spatiotemporal chaos characteristic of this equation and replace it with near perfect spatial periodicity. When an elemental material is bombarded with non-volatile ions, hexagonal arrays of nanoholes, highly ordered ripples and herringbone patterns can emerge. In the final part of the talk, a mathematical model that can account for all of these experimental observations will be discussed.