Plant neocentromeres are large heterochromatic domains that associate with microtubules and move rapidly poleward during meiotic cell division. In maize, neocentromeres are part of a process that leads to the preferential recovery (meiotic drive) of knobs in progeny. These 'classical' plant neocentromeres differ from animal neocentromeres by their morphology, inability to mediate sister chromatid cohesion, and their rates of movement on the spindle. We provide a comprehensive review of classical neocentromeres with emphasis on their origin and mechanisms of motility. The data support the view that most, if not all, classical neocentromeres are the outcome of selection by meiotic drive. In addition, we compare and contrast neocentromere-mediated meiotic drive with a recently proposed meiotic drive model for centromere evolution.