This chapter aims to (1) provide background for conceptual mathematical models of spontaneous pattern formation, in the context of dryland vegetation patterns, and (2) review observational studies of the phenomenon. The chapter also highlights challenges and opportunities associated with the development of the models in light of increasing availability of remote sensing data. This includes both satellite imagery of the patterns and elevation data of the topography. The vast scales, in time and space, associated with the key processes further suggest avenues for improved mathematical modeling paradigms.