A beautiful example of spontaneous pattern formation occurs in certain dryland environments around the globe, where stripes of vegetation alternate with stripes of bare soil, with striking regularity and on a scale readily monitored by remote-sensing satellites. Theoretical ecologists, together with theoretical physicists and applied mathematicians, have also exploited the physicist's understanding of mechanisms of spontaneous self-organization to model these dryland vegetation patterns. They have shown how these spatial patterns exploit positive feedbacks to efficiently concentrate the limited water resource into vegetated zones. This talk will present a few research vignettes on this topic of vegetation pattern formation in dryland ecosystems. It is research that is motivated by the vulnerability of these fragile ecosystems to increased desertification under climate change and changes in human land-use.