Electron and energy transfer processes between an atom or molecule and a surface are extremely important for many applications in physics and chemistry. Therefore a profound understanding of these processes is essential in order to analyze a large variety of physical systems. The microscopic description of the two-electron Auger processes, leading to neutralization/ionization of an ion/neutral atom in front of a solid surface, has been a long-standing problem. It can be dated back to the 1950s when H.D. Hagstrum proposed to use the information contained in the spectrum of the electrons emitted during the neutralization of slow noble gas ions as a surface analytical tool complementing photoelectron spectroscopy. However, only recently a comprehensive description of the Auger neutralization mechanism has been achieved by the combined efforts of theoretical and experimental methods. In this article we review the theoretical models for this problem, stressing how their outcome compare with experimental results. We also analyze the inverse problem of Auger ionization. We emphasize the understanding of the key quantities governing the processes and outline the challenges remaining. This opens new perspectives for future developments of theoretical and experimental work in this field.