Viruses vastly outnumber their host cells and must present a huge selective pressure. It is also becoming evident that only a small percent of the eukaryotic genome codes for molecules involved in cellular structures and functions, and that much of the remainder may have a viral origin. Viruses clearly play a central role in the biosphere, but how is this viral world organized? Classification was originally based on virus morphology and the particular host infected, but now there is an increasing trend to rely on sequence information. The type of genome (e.g., RNA or DNA, single- or double-stranded) provides fundamental classification criteria, while sequence comparisons can provide fine mapping for closely related viruses. However, it is currently very difficult to identify long-range evolutionary relationships. We present here a different approach, based on the idea that each virus has an innate "self." When the structures and functions characteristic of this "self" are identified, then they uncover relationships beyond those accessible from sequence information alone. The new approach is illustrated by sketching some possible viral lineages. We propose that urviruses were present before the division of cellular life into its current domains, and that the viral world has lineages that can be traced back to the root of the universal tree of life.