Patterns in species incidence and compositional turnover are central to understanding what drives biodiversity. Here we propose zeta (ζ) diversity, the number of species shared by multiple assemblages, as a concept and metric that unifies incidence-based diversity measures, patterns, and relationships. Unlike other measures of species compositional turnover, zeta diversity partitioning quantifies the complete set of diversity components for multiple assemblages, comprehensively representing the spatial structure of multispecies distributions. To illustrate the application and ecological value of zeta diversity, we show how it scales with sample number, grain, and distance. Zeta diversity reconciles several different biodiversity patterns, including the species accumulation curve, the species-area relationship, multispecies occupancy patterns, and scaling of species endemism. Exponential and power-law forms of zeta diversity are associated with stochastic versus niche assembly processes. Zeta diversity may provide new insights on biodiversity patterns, the processes driving them, and their response to environmental change.