This review discusses some of the history of the fundamental, but not fully solved problem of how the centrosome duplicates from one to only two as the cell prepares for mitosis. We start with some of the early descriptions of the centrosome and the remarkably prescient but then controversial inferences drawn concerning its function in the cell. For more than 100 years, one of the most difficult issues for the concept of the centrosome has been to integrate observations that centrosomes appear to be important for spindle assembly in animal cells yet are not evident in higher plant cells and some animal cells. This stirred debate over the existence of centrosomes and their importance. A parallel debate concerned the role of the centrioles in organizing centrosomes. The relatively recent elucidation of bipolar spindle assembly around chromatin allows a re-examination of the role of centrioles in controlling centrosome duplication in animal cells. The problem of how centrosomes precisely double in preparation for mitosis in animal cells has now moved to the mystery of how only one procentriole is assembled at each mother centriole.