The bombardment of surfaces by ions can lead to the spontaneous formation of nano-structures. Depending on the irradiation conditions, smoothening or roughening mechanisms can be the leading order in pattern formation which can result in the creation of dots, ripples or ultra-smoothening effects. Because ion bombardment is already ubiquitous in industrial settings, and is relatively inexpensive compared to other surface processing techniques, self-organized patterning by ion bombardment could enable a simple, economical means of inducing well-defined nanoscale structures in a variety of settings. Understanding the fundamental behavior of surfaces during ion bombardment is therefore a vital goal; however, a complete understanding of physical processes governing surface pattern formation has not been reached yet. In order to address this issue, my thesis research has utilized three primary approaches. First, I have done real-time non-coherent X-ray scattering experiments at Cornell High Energy Synchrotron Source (CHESS) for studying kinetics of structure formation of Silicon undergoing Ar⁺ bombardment over a range of wavenumbers 4-5 times larger than has previously been obtained. From our data, we were able to extract values of the angle-dependent thickness of the amorphous layer that forms under ion bombardment, the ion-enhanced fluidity within that film, the magnitude of the stress being generated by the ion beam, and the strength of prompt atomic displacement mechanisms. Second, to further deepen our knowledge of surface dynamics, I have performed coherent X-ray studies of Ar⁺ bombardment of SiO₂ at the Advanced Photon Source (APS) for investigating the dynamics more profoundly than can be done with traditional time-resolved experiments. When using a focused ion beam, an inhomogeneous ripple motion was generated, this phenomenon reflected as an oscillatory behavior in the two-time and corresponding g₂(t) correlation functions. By fitting the oscillations in the g₂(t) correlation function, we have determined the surface ripple velocity on SiO₂ driven by Ar⁺ sputter erosion. Finally, to support the results of coherent X-ray experiments, simulations of growth models such as linear Kuramoto-Sivashinsky (KS) and Kardar-Parisi-Zhang (KPZ) have been carried out in order to compare the simulated temporal correlation functions of the scattered intensity with those obtained from the coherent x-ray scattering experiments.