Vegetation in many semiarid regions is strikingly patterned, forming regular stripes on hillsides and irregular mosaics on flat ground. A simple model of plant and water dynamics based on ecologically realistic assumptions and with reasonable parameter values captures both of these types of patterns. The regular patterns result from a Turing-like instability; the irregular patterns arise when the ecological dynamics amplify slight small-scale topographic variability. Because of the close agreement between observations and these theoretical results, this system provides a clear example of how nonlinear mechanisms can be important in determining the spatial structure of plant communities.