Spatially defined arrays of droplets differ from bulk emulsions in that droplets in arrays can be indexed on the basis of one or more spatial variables to enable identification, monitoring, and addressability of individual droplets. Spatial indexing is critical in experiments with hundreds to millions of unique compartmentalized microscale processes--for example, in applications such as digital measurements of rare events in a large sample, high-throughput time-lapse studies of the contents of individual droplets, and controlled droplet-droplet interactions. This review describes approaches for spatially organizing and manipulating droplets in one-, two-, and three-dimensional structured arrays, including aspiration, laminar flow, droplet traps, the SlipChip, self-assembly, and optical or electrical fields. This review also presents techniques to analyze droplets in arrays and applications of spatially defined arrays, including time-lapse studies of chemical, enzymatic, and cellular processes, as well as further opportunities in chemical, biological, and engineering sciences, including perturbation/response experiments and personal and point-of-care diagnostics.