In this brief account we specifically address the question of how the plasma membrane-associated basal body/axoneme of the unicellular ancestor of eukaryotes has evolved into the centrosome organelle through the several attempts to multicellularity. We propose that the connection between the flagellar apparatus and the nucleus has been a critical feature for leading to the centriole-based centrosome of metazoa, the Spindle Pole Body of fungi, or to the absence of any centrosome in seed plants. We further suggest that the evolution of this connection could be reflected in the evolution of the centrin proteins. We then review evidence showing that the evolution of the centrosome-based tubulin network has been correlated with the evolution of the cortical actin-based cleavage apparatus. Finally we argue that this coevolution had a major impact on the cell individuation process and on the evolution of multicellular organisms. We conclude that only the metazoan lineage evolved multicellularity without loosing the ancestral association of three basic cellular functions of the basal body/axoneme or the derived centrosome organelle, namely sensation, motion and division.