A key question in invasion biology is why some regions have more alien species than others. Here, we provide a general framework to answer this. We model alien species richness as a function of the number of species introduced (colonization pressure) and the probability that each species establishes, which is a function of propagule pressure (the number of introduction events and the number of individuals per event) and the probability that a founding individual leaves a surviving lineage (lineage survival probability). With this model we show that: (a) alien species richness is most sensitive to variation in colonization pressure; (b) heterogeneity in lineage survival probability lowers the probability of population establishment, implying alien richness should be lower in more spatially or temporally variable environments; (c) heterogeneity in lineage survival probability leads to higher alien richness when a given propagule pressure is divided into more introduction events, each involving fewer individuals; and (d) we cannot quantify how specific components, such as lineage survival probability, influence alien species richness without data on other components and knowledge of how these covary. Overall, the model provides novel insights into the factors influencing alien species richness, and shows why we expect anthropogenic effects to be critical to this.