This review will first recall the phenomena of "cortical inheritance" observed and genetically demonstrated in Paramecium 40 years ago, and later in other ciliates (Tetrahymena, Oxytricha, Paraurostyla), and will analyze the deduced concept of "cytotaxis" or "structural memory." The significance of these phenomena, all related (but not strictly restricted) to the properties of ciliary basal bodies and their mode of duplication, will be interpreted in the light of present knowledge on the mechanism and control of basal body/centriole duplication. Then other phenomena described in a variety of organisms will be analyzed or mentioned which show the relevance of the concept of cytotaxis to other cellular processes, mainly (1) cytoskeleton assembly and organization with examples on ciliates, trypanosome, mammalian cells and plants, and (2) transmission of polarities with examples on yeast, trypanosome and metazoa. Finally, I will discuss some aspects of this particular type of non-DNA inheritance: (1) why so few documented examples if structural memory is a basic parameter in cell heredity, and (2) how are these phenomena (which all rely on protein/protein interactions, and imply a formatting role of preexisting proteinic complexes on neo-formed proteins and their assembly) related to prions?