Ion bombardment at extreme grazing incidence leads to the formation of remarkably well ordered, one to two atomic layer deep, parallel grooves on a Cu(001) surface. These self-organized grooves are oriented parallel to the ion's plane of incidence and their period can be controlled between ∼4 and 15 nm. We have identified two distinct temperature regimes: Between about 200 and 300 K the substrate temperature controls the period, while below 200 to about 150 K the ion energy does. The groove separation distribution shows distinct self-affine character. The physical origin of the novel nanostructures is discussed.