The chromosomes of mammals tend to be either mostly acrocentric (having one long arm) or mostly bi–armed, with few species having intermediate karyotypes. The theory of centromeric drive suggests that this observation reflects a bias during female meiosis, favouring either more centromeres or fewer, and that the direction of this bias changes frequently over evolutionary time. B chromosomes are selfish genetic elements found in some individuals within some species. B chromosomes are often harmful, but persist because they drive (i.e. they are transmitted more frequently than expected). We predicted that species with mainly acrocentric chromosomes would be more likely to harbour B chromosomes than those with mainly bi–armed chromosomes, because female meiosis would favour more centromeres over fewer in species with one–armed chromosomes. Our results show that B chromosomes are indeed more common in species with acrocentric chromosomes, across all mammals, among rodents, among non–rodents and in a test of independent taxonomic contrasts. These results provide independent evidence supporting the theory of centromeric drive and also help to explain the distribution of selfish DNA across species. In addition, we demonstrate an association between the shape of the B chromosomes and the shape of the typical (‘A’) chromosomes.