Neocentromeres are rare human chromosomal aberrations where a new centromere has formed in a previously non-centromeric location. The emergence of new centromeres on a chromosome that already contains an endogenous centromere would be a highly deleterious event which would lead to dicentricity and mitotic instability. Nonetheless, neocentromere formation appears to provide a mechanism for the acquisition of a new centromere. Neocentromeres are most often observed on chromosomal arm fragments that have separated from an endogenous centromere, and therefore actually lead to mitotic stability of what would have been an acentric fragment. Neocentromeres have recently also been observed on apparently unrearranged chromosomes where the endogenous centromere has been inactivated. Furthermore, the process of centromere repositioning during primate chromosomal evolution may depend on the acquisition and subsequent fixation of neocentromeres. This remarkable plasticity in the position of centromeres has important implications for human cytogenetics and chromosome evolution, and provides an opportunity to further our understanding of the process of centromere formation and structure.