The extension of virology beyond its traditional medical, veterinary or agricultural applications, now called environmental virology, has shown that viruses are both the most numerous and diverse biological entities on earth. In particular, virus isolation studies involving unicellular eukaryotic hosts (heterotrophic and photosynthetic protozoans) revealed numerous viral types previously unexpected in terms of virion structure and morphology, genome size and gene content, or mode of replication. Complemented by large-scale metagenomic analyzes, these discoveries have rekindled interest in the enigma of the evolutionary origin of viruses, for which no simple definition encompassing all of their diversity is still unanimous. Several laboratories have repeatedly tackled the deep reconstruction of the evolutionary history of viruses, using various methods of molecular phylogeny applied to the few shared genes detected in certain virus groups (e.g. the Nucleocytoviricota). Beyond the practical difficulties of establishing reliable homology relationships from extremely divergent sequences, I present here purely conceptual arguments highlighting several fundamental limitations plaguing the reconstruction of the deep evolutionary history of viruses, and even more the identification of their unique of multiple origin (s). Those limitations are direct consequences of the particularly random mechanisms which govern the reductive evolution of obligate intracellular parasites.