An overview of edge and divertor physics research on ASDEX Upgrade of relevance for next-step fusion devices like ITER is presented. The results described were primarily obtained in lower single-null divertor configurations with three consecutive bottom divertor designs, starting from an initial open divertor (Div I) over the closed LYRA configuration (Div II), optimized for low-triangularity single-null equilibria, to the presently operational variant Div IIb, fitting a large variety of plasma shapes. The upper, geometrically open divertor structure remained essentially unchanged. A dedicated diagnostics system in combination with advanced plasma control scenarios and extensive numerical modeling allowed for a detailed analysis of edge and divertor physics mechanisms. Main chamber edge profiles exhibit a double structure, especially pronounced in high-performance H-mode plasmas. While radial transport inside and across the separatrix is governed by critical gradients, the cold scrape-off layer wing shows rapid diffusion or even outward drift, probably related to intermittent crossfield transport. The divertor behavior has been studied for the different divertor geometries and for all operational regimes of interest. Closed divertor operation enhances divertor recycling and pumping, reduces the power load on target plates by increased upstream losses, and facilitates onset of plasma detachment. The transient power load during type I ELMs, however, remains high and problematic, while the small type III ELMs, appearing, for example, in radiative discharge scenarios, and especially the type II ELMs are nearly invisible on the target heat flux. Despite this strong effect of divertor geometry on the divertor behavior, its direct effect on core confinement remains small.