  This is a book chapter, written as a contribution to a new edition of
"Dryland Ecohydrology", edited by Paolo D'Odorico, Amilcare Porporato, and
Christiane Runyan, (to appear, Springer 2019). It aims to (1) describe some of
the background to conceptual mathematical models of spontaneous pattern
formation, in the context of dryland vegetation patterns, and (2) review some
of the 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.
