I review the mathematical and biological aspects of Hubbell's (2001) neutral theory of species abundance for ecological communities, and clarify its historical connections with closely related approaches in population genetics. A selective overview of the empirical evidence for and against this theory is provided, with a special emphasis on tropical plant communities. The neutral theory predicts many of the basic patterns of biodiversity, confirming its heuristic power. The strict assumption of equivalence that defines neutrality, equivalence among individuals, finds little empirical support in general. However, a weaker assumption holds for stable communities, the equivalence of average fitness among species. One reason for the surprising success of the neutral theory is that all the theories of species coexistence satisfying the fitness equivalence assumption, including many theories of niche differentiation, generate exactly the same patterns as the neutral theory. Hubbell's neutral theory represents an important synthesis and a much needed demonstration of the pivotal role of intraspecific variability in ecology. Further improvements should lead to an explicit linking to niche-based processes. This research programme will depend crucially on forthcoming theoretical and empirical achievements.