The centrosome (centriole) and the cytoskeleton produced by it are structures, which probably determine differentiation, morphogenesis, and switching on the mechanism of replicative aging in all somatic cells of multicellular animals. The mechanism of such programming of the events seems to include cytoskeleton influences and small RNAs related to the centrosome. 1) If these functions are really related with centrioles, the multicellular organism's cells which: a) initially lack centrioles (e.g., higher plant cells and also zygote and early blastomeres of some animals) or cytoskeleton (e.g., embryonic stem cells); or b) generate centrioles de novo (e.g., zygote and early blastomeres of some animals), will be totipotent and lack replicative aging. Consequently, the absence (constant or temporary) of the structure determining the counting of divisions also means the absence of counting of differentiation processes. 2) Although a particular damage to centrioles or cytoskeleton (e.g., in tumor cells) fails to make the cells totipotent (because the morphogenetic status of these cells, as differentiated from that of totipotent ones, is not zero), but such a transformation can suppress the initiation of the aging mechanism induced by these structures and, thus, make such cells replicatively "immortal".