Viruses are ubiquitous and abundant pathogens that likely infect and disrupt all cellular organisms. Although a high diversity of viral types has been characterized from corals over the last decade, we are only beginning to understand the distribution, dynamics, and roles of viral consortia in reef ecosystems. This review synthesizes previous work on the viruses associated with corals and their cnidarian relatives, and highlights some strengths and limitations of the methods commonly used for viral analysis. Based on these foundational studies, we present a current working model for stony coral virology. We hypothesize that viruses are present on and within corals and that these consortia commonly include representatives of the Herpesviridae, Phycodnaviridae, and phages and archaeal viruses. Our working model posits that these different viral groups target distinct hosts, specifically the coral itself, its endosymbiotic algae (Symbiodinium spp.), and its resident microbiome, respectively. Corals may also harbor additional viral diversity (e.g., members of the Iridoviridae) that is pathogenic to reef-associated organisms. Viral infections likely underlie some coral diseases and may be responsible for reef epizootics, but evidence for viral-mediated coral disease is currently inconclusive. Future efforts in coral viral ecology should focus on determining if and how viral infection contributes to reduced coral fitness and/or benthic community shifts in reef ecosystems.