Arbuscular mycorrhizal fungi in the phylum Glomeromycota are found globally in most vegetation types, where they form a mutualistic symbiosis with plant roots. Despite their wide distribution, only relatively few species are described. The taxonomy is based on morphological characters of the asexual resting spores, but molecular approaches to community ecology have revealed a considerable unknown diversity from colonized roots. Although the lack of genetic recombination is not unique in the fungal kingdom, arbuscular mycorrhizal fungi are probably ancient asexuals. The long asexual evolution of the fungi has resulted in considerable genetic diversity within morphologically recognizable species, and challenges our concepts of individuals and populations. This review critically examines the concepts of species, communities, populations and individuals of arbuscular mycorrhizal fungi.