Self-organized, nano-scale structures appear on solid surfaces under ion beam irradiation with ion energies in the keV range. Within the last decade, surface engineering by ion beam sputtering (IBS) has become a very promising candidate for bottom-up production techniques of nano-devices. Morphologies like ripples, and regular arrays of dots, pyramids and pits as well as ultra-smooth surfaces have been obtained on a wide variety of substrates, including important semiconductor materials like Si, Ge, GaAs and InP. In spite of many substantial improvements of experimental surface structuring by IBS, the physical mechanisms underlying the pattern formation are still poorly understood. In this book we report the results obtained by usage of a Kinetic Monte Carlo simulations and continuum theory to study the physical mechanisms in atomic scale, which lead to pattern formation in general, as well as further possibilities to improve and fine-tune the properties of emerging patterns.