<p>The overall complexity of organisms increases during the course of evolution, starting with the first self-replicating molecules, followed by prokaryotes and eukaryotes, first unicellular and later multicellular. We present an opinion that non-cellular genetic entities such as transposable elements, plasmids, viruses and viroids, although originally parasitic, selfish and sometimes destructive elements, may contribute to the increase of complexity. We propose that non-cellular genetic elements impose (parasitic) pressure on the cooperative genes of cellular organisms, driving the sequence of evolutionary transitions from the first cooperative replicators to multicellular life forms, and have suggested that they belong to a separate kingdom of life, the Acytota. The complexity increase is probably caused by the high proliferation capacity of these non-cellular genetic elements, their frequent horizontal gene transfer, participation in parasite-host arms races, formation of epigenetic silencing mechanisms as well as the ability to build genetic regulatory networks. Simultaneously, these elements contribute to complexity by supplying genetic material via domestication, genome rearrangements, and dispersal of regulatory elements. Complexity has not only increased during evolution, there are also examples of simplification, both during chemical evolution (in prebiotic chemistry) and the evolution of parasites. Therefore, we describe the ups and downs of organism complexity and discuss the reasons for the general dominant upward trend, namely coevolution and the interaction of existing modules.</p>