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Abstract Species richness (the number of species) in an assemblage is a key metric in many research fields of ecology. Simple counts of species in samples typically underestimate the true species richness and strongly depend on sampling effort and sample completeness. Based on possibly unequal‐sampling effort and incomplete samples that miss many species, there are two approaches to infer species richness and make fair comparisons among multiple assemblages,: (1) An asymptotic approach via species richness estimation. This approach aims to compare species richness estimates across assemblages. We focus on the nonparametric estimators that are universally valid for all species abundance distributions. (2) A non‐asymptotic approach via the sample‐size‐ and coverage‐based rarefaction and extrapolation on the basis of standardised sample size or sample completeness (as measured by sample coverage). This approach aims to compare species richness estimates for equally large or equally complete samples. Two R packages (SpadeR and iNEXT) are applied to beetle data for illustration. Key Concepts Due to sampling limitation, there are undetected species in almost every biodiversity survey. Empirical species counts underestimate species richness and highly depend on sampling efforts and sample completeness. Based on incomplete samples, species richness (observed plus undetected) is statistically difficult to estimate accurately especially for highly diverse assemblages with many rare species. Abundant species (which are certain to be detected in samples) contain almost no information about the undetected species richness. Rare species (which are likely to be either undetected or infrequently detected) contain nearly all the information about the undetected species richness. Most nonparametric estimators of the number of undetected species are based on the frequency counts of the detected rare species, e.g. singletons and doubletons for abundance data. Nonparametric estimators of species richness are universally valid for all species abundance distributions and thus are more robust than parametric estimators that are based on specified parametric abundance models. Rarefaction and extrapolation methods allow for fair and meaningful comparison of species richness estimates for standardised samples on the basis of sample size or sample completeness. Sample‐size‐based rarefaction and extrapolation methods aim to compare species richness estimates for equally large samples determined by samplers. Coverage‐based rarefaction and extrapolation methods aim to compare species richness estimates for equally complete samples or equal fractions of population individuals reliably estimated from data.