Modellers of large scale genome rearrangement events, in which segments of DNA are inverted, moved, swapped, or even inserted or deleted, have found a natural syntax in the language of permutations. Despite this, there has been a wide range of modelling choices, assumptions and interpretations that make navigating the literature a significant challenge. Indeed, even authors of papers that use permutations to model genome rearrangement can struggle to interpret each others' work, because of subtle differences in basic assumptions that are often deeply ingrained (and consequently sometimes not even mentioned). In this paper, we describe the different ways in which permutations have been used to model genomes and genome rearrangement events, presenting some features and limitations of each approach, and show how the various models are related. This paper will help researchers navigate the landscape of genome rearrangement models, and make it easier for authors to present clear and consistent models.