<p>The species problem dates to antiquity and encompasses three aspects since the modern synthesis: what species are, how they originate, and how they are delimited. Species are frequently thought of as evolutionary entities with real but fuzzy boundaries in space and time, but this concept struggles to constrain groups such as prokaryotes or lichens that push conventional notions of genealogy. Speciation is increasingly well understood, but questions remain regarding the origin, prevalence, and strength of reproductive isolation and genomic divergence of lineages. Delimitation is facilitated by sophisticated algorithms, but additional axes such as geography, ecology, and phenotype should be sampled, with speciation hypotheses tying those attributes to evolutionary processes. In this Review, we survey these topics, outline major open questions, and identify two new dimensions of the species problem: how these three dimensions are altered in the Anthropocene, and whether species have intrinsic value in themselves. Global change alters the intensity and trajectory of natural processes such as hybridization, and introduces new niches through urbanization and domestication, and new species are even being created in the laboratory. In the realm of conservation, it is unclear whether species stand apart from other facets of biodiversity in having a unique intrinsic value of their own.</p>