A prime aim of invasion biology is to predict which species will become invasive, but retrospective analyses have so far failed to develop robust generalizations. This is because many biological, environmental, and anthropogenic factors interact to determine the distribution of invasive species. However, in this paper we also argue that many analyses of invasiveness have been flawed by not considering several fundamental issues: (1) the range size of an invasive species depends on how much time it has had to spread (its residence time); (2) the range size and spread rate are mediated by the total extent of suitable (i.e. potentially invasible) habitat; and (3) the range size and spread rate depend on the frequency and intensity of introductions (propagule pressure), the position of founder populations in relation to the potential range, and the spatial distribution of the potential range. We explored these considerations using a large set of invasive alien plant species in South Africa for which accurate distribution data and other relevant information were available. Species introduced earlier and those with larger potential ranges had larger current range sizes, but we found no significant effect of the spatial distribution of potential ranges on current range sizes, and data on propagule pressure were largely unavailable. However, crucially, we showed that: (1) including residence time and potential range always significantly increases the explanatory power of the models; and (2) residence time and potential range can affect which factors emerge as significant determinants of invasiveness. Therefore, analyses not including potential range and residence time can come to misleading conclusions. When these factors were taken into account, we found that nitrogen-fixing plants and plants invading arid regions have spread faster than other species, but these results were phylogenetically constrained. We also show that, when analysed in the context of residence time and potential range, variation in range size among invasive species is implicitly due to variation in spread rates, and, that by explicitly assuming a particular model of spread, it is possible to estimate changes in the rates of plant invasions through time. We believe that invasion biology can develop generalizations that are useful for management, but only in the context of a suitable null model.