Species diversity is the number of species present in an area or sample, together with the characteristics of their relative abundance distribution, especially its evenness. Species diversity is only one dimension of biodiversity, but because species are fundamental units of ecology and evolution, and because most diversity data consist of species presence/absence, species diversity is often central to assessing and conserving biodiversity. After noting major conceptual and practical issues with species diversity and species richness, we discuss how best to estimate them from samples both for estimating total species number and for comparing diversity across samples and areas. Despite errors associated with extrapolation, recent research has demonstrated that in many cases, given adequate sampling, both nonparametric and parametric methods can estimate total species richness and effectively standardize across samples. Much research has focused on developing and interpreting compound diversity measures that incorporate patterns of both richness and abundance, some with weightings for additional factors such as rarity, phylogenetic distinctness, and ecological function. For a well‐known set of these indices, including the Shannon and Simpson indices, lognormal mean and variance, log‐series α and related neutral theory parameters, we discuss their consistency in ordering samples and their sensitivity and dependence on sample size. Finally, we touch upon major global patterns of diversity including estimates of the number of species on earth (5 million to 50 million) and large‐scale diversity gradients.