Significance Invasive alien species pose major threats to biodiversity and ecosystems. However, identifying drivers of invasion success has been challenging, in part because species can achieve invasiveness in different ways, each corresponding to different aspects of demographics and distribution. Employing a multidimensional perspective of invasiveness to Europe’s alien flora, we find species generally fall along an axis from overall poor invaders to super invaders that become abundant, widespread, and invade diverse habitats. Some species that deviate from this pattern are recently introduced and still spreading, but others represent atypical forms of invasiveness. In addition to identifying species traits and ecological circumstances associated with super invaders (e.g., intercontinental introductions), we explore drivers in atypical invasions, providing increased clarity into invasion processes.