Understanding drivers of success for alien species can inform on potential future invasions. Recent conceptual advances highlight that species may achieve invasiveness via performance along at least three distinct dimensions: 1) local abundance, 2) geographic range size, and 3) habitat breadth in naturalized distributions. Associations among these dimensions and the factors that determine success in each have yet to be assessed at large geographic scales. Here, we combine data from over one million vegetation plots covering the extent of Europe and its habitat diversity with databases on species' distributions, traits, and historical origins to provide a comprehensive assessment of invasiveness dimensions for the European alien seed plant flora. Invasiveness dimensions are linked in alien distributions, leading to a continuum from overall poor invaders to super invaders - abundant, widespread aliens that invade diverse habitats. This pattern echoes relationships among analogous dimensions measured for native European species. Success along invasiveness dimensions was associated with details of alien species' introduction histories: earlier introduction dates were positively associated with all three dimensions, and consistent with theory-based expectations, species originating from other continents, particularly acquisitive growth strategists, were among the most successful invaders in Europe. Despite general correlations among invasiveness dimensions, we identified habitats and traits associated with atypical patterns of success in only one or two dimensions - for example, the role of disturbed habitats in facilitating widespread specialists. We conclude that considering invasiveness within a multidimensional framework can provide insights into invasion processes while also informing general understanding of the dynamics of species distributions.