Viruses were defined as one of the two principal types of organisms in the biosphere, namely, as capsid-encoding organisms in contrast to ribosome-encoding organisms, i.e., all cellular life forms. Structurally similar, apparently homologous capsids are present in a huge variety of icosahedral viruses that infect bacteria, archaea, and eukaryotes. These findings prompted the concept of the capsid as the virus “self” that defines the identity of deep, ancient viral lineages. However, several other widespread viral “hallmark genes” encode key components of the viral replication apparatus (such as polymerases and helicases) and combine with different capsid proteins, given the inherently modular character of viral evolution. Furthermore, diverse, widespread, capsidless selfish genetic elements, such as plasmids and various types of transposons, share hallmark genes with viruses. Viruses appear to have evolved from capsidless selfish elements, and vice versa, on multiple occasions during evolution. At the earliest, precellular stage of life's evolution, capsidless genetic parasites most likely emerged first and subsequently gave rise to different classes of viruses. In this review, we develop the concept of a greater virus world which forms an evolutionary network that is held together by shared conserved genes and includes both bona fide capsid-encoding viruses and different classes of capsidless replicons. Theoretical studies indicate that selfish replicons (genetic parasites) inevitably emerge in any sufficiently complex evolving ensemble of replicators. Therefore, the key signature of the greater virus world is not the presence of a capsid but rather genetic, informational parasitism itself, i.e., various degrees of reliance on the information processing systems of the host.