Asexuality is a derived character, and asexual species have emerged multiple times independently throughout evolution. In animals, parthenogenesis, whereby a zygote inherits only its mother genome, requires female meiosis modification to produce unreduced gametes and compensate for the lack of outcrossing. Various changes in the meiotic steps have been identified. Here, we propose a simplified nomenclature when discussing asexuality in animals, as the existing terminology has caused many misinterpretations and may impede interdisciplinary collaboration. We also shed light on the constraints related to these meiotic modifications in asexual animals and their consequences on their genome. When recombination is preserved in parthenogens, most meiotic modifications should, according to Mendel's second law, result in a loss of heterozygosity (LOH). On the other hand, LOH can be prevented if recombination is either abolished or strongly reduced during meiotic prophase. Surprisingly, several recent examples have demonstrated that LOH can be prevented despite recombination. We will describe these examples in detail as they illustrate the constraints underlying the origin and evolution of asexuality. We also emphasize that the majority of mechanisms remain unknown in many species. This highlights the need to revisit the old literature using state-of-the-art cytological imaging and genomic techniques to shed light on these unexplored processes.