Virophages are small dsDNA viruses that hijack the machinery of giant viruses during the co-infection of a protist (i.e., microeukaryotic) host and represent an exceptional case of "hyperparasitism" in the viral world. While only a handful of virophages have been isolated, a vast diversity of virophage-like sequences have been uncovered from diverse metagenomes. Their wide ecological distribution, idiosyncratic infection and replication strategy, ability to integrate into protist and giant virus genomes and potential role in antiviral defense have made virophages a topic of broad interest. However, one limitation for further studies is the lack of clarity regarding the nomenclature and taxonomy of this group of viruses. Specifically, virophages have been linked in the literature to other "virophage-like" mobile genetic elements and viruses, including polinton-like viruses (PLVs), but there are no formal demarcation criteria and proper nomenclature for either group, i.e., virophage or PLVs. Here, as part of the ICTV Virophage Study Group, we leverage a large set of genomes gathered from published datasets as well as newly generated protist genomes to propose delineation criteria and classification methods at multiple taxonomic ranks for virophages 'sensu stricto', i.e., genomes related to the prototype isolates Sputnik and mavirus. Based on a combination of comparative genomics and phylogenetic analyses, we show that this group of virophages forms a cohesive taxon that we propose to establish at the class level and suggest a subdivision into four orders and seven families with distinctive ecogenomic features. Finally, to illustrate how the proposed delineation criteria and classification method would be used, we apply these to two recently published datasets, which we show include both virophages and other virophage-related elements. Overall, we see this proposed classification as a necessary first step to provide a robust taxonomic framework in this area of the virosphere, which will need to be expanded in the future to cover other virophage-related viruses such as PLVs.