The original theory of the centrosome as the 'reproductive organ' of the cell provides a logical explanation of the mitotic poles and the accuracy of cell division. No alternative explanation has replaced it. The historical problem was the failure to identify centrosomes as compact physical bodies in a great many kinds of cells. In this essay, I consider the evidence that centrosomes are flexible bodies; they may take on alternative forms and their forms determine the shapes of mitotic poles and other organizers of microtubular structures. Compact corpuscular centrosomes are not necessary and would not be expected in cases where microtubules clearly do not originate from point sources. A model of the flexible centrosome is introduced and the speculation that the centrosome is a bearer of morphological information is considered.