Dicentric chromosomes have been identified as instigators of the genome instability associated with cancer, but this instability is often resolved by one of a number of different secondary events. These include centromere inactivation, inversion, and intercentromeric deletion. Deletion or excision of one of the centromeres may be a significant occurrence in myeloid malignancy and other malignancies but has not previously been widely recognized, and our reports are the first describing centromere deletion in cancer cells. We review what is known about dicentric chromosomes and the mechanisms by which they can undergo stabilization in both constitutional and cancer genomes. The failure to identify centromere deletion in cancer cells until recently can be partly explained by the standard approaches to routine diagnostic cancer genome analysis, which do not identify centromeres in the context of chromosome organization. This hitherto hidden group of primary dicentric, secondary monocentric chromosomes, together with other unrecognized dicentric chromosomes, points to a greater role for dicentric chromosomes in cancer initiation and progression than is generally acknowledged. We present a model that predicts and explains a significant role for dicentric chromosomes in the formation of unbalanced translocations in malignancy.