Mitochondrial DNA (mtDNA) is typically passed on to progeny only by the female parent. The phenomenon of "doubly uniparental inheritance" (DUI) of mtDNA in many bivalve species is a fascinating exception to the paradigm of strict maternal inheritance of mtDNA. In this review, we survey the current state of knowledge of DUI, and discuss several active areas of research in this field. Topics/questions covered include: the number of times DUI evolved (once or multiple origins), the link between DUI and sex determination, the role(s) of mtDNA-encoded non-oxidative phosphorylation genes (i.e., ORFan/orf genes) in freshwater mussels, the function of conserved sequence motifs and sperm transmission elements in mtDNA of marine mussels, the challenges of annotating mtDNA genomes of DUI species, the presence of unorthodox features in venerid mtDNA, whether or not orf DNA sequences are useful in species-level identification of freshwater mussel, and finally, whether or not there are obvious benefits of DUI. For each topic we also highlight important avenues for future research within this fascinating field of mitochondrial evolutionary biology.