The identification of CENPA, CENPB, and CENPC by Earnshaw and Rothfield 40 years ago has revealed the remarkable diversity and complexity of centromeres and confirmed most seed plants and animals have centromeres comprised of complex satellite arrays. The rapid evolution of centromeres and positive selection on CENPA and CENPC led to the centromere drive model, in which competition between tandem satellite arrays of differing size and centromere strength for inclusion in the egg of animals or megaspore of seed plants during female meiosis drives rapid evolution of centromeres and kinetochore proteins. Here we review recent work showing that non-B-form DNA structures in satellite centromeres make them sites of frequent replication fork stalling, and that repair of collapsed forks by break-induced replication rather than unequal sister chromatid exchange is likely the primary mode of satellite expansion and contraction, providing the variation in satellite copy number that is the raw material of centromere drive. Centromere breaks at replication, rather than errors at mitosis, can account for most centromere misdivisions that underlie aneuploidies in cancer.