Viruses infect members of domains Bacteria, Eukarya, and Archaea. While those infecting domain Eukarya are nearly universally described as "Viruses", those of domain Bacteria, to a substantial extent, instead are called "Bacteriophages," or "Phages." Should the viruses of domain Archaea therefore be dubbed "Archaeal phages," "Archaeal viruses," or some other construct? Here we provide documentation of published, general descriptors of the viruses of domain Archaea. Though at first the term "Phage" or equivalent was used almost exclusively in the archaeal virus literature, there has been a nearly 30-year trend away from this usage, with some persistence of "Phage" to describe "Head-and-tail" archaeal viruses, "Halophage" to describe viruses of halophilic Archaea, use of "Prophage" rather than "Provirus," and so forth. We speculate on the root of the early 1980's transition from "Phage" to "Virus" to describe these infectious agents, consider the timing of introduction of "Archaeal virus" (which can be viewed as analogous to "Bacterial virus"), identify numerous proposed alternatives to "Archaeal virus," and also provide discussion of the general merits of the term, "Phage." Altogether we identify in excess of one dozen variations on how the viruses of domain Archaea are described, and document the timing of both their introduction and use.