1. Invasive alien species are one of the major threats to global biodiversity, ecosystem integrity, nature's contributions to people and human health. While scenarios about potential future developments have been available for other global change drivers for quite some time, we largely lack an understanding of how biological invasions might unfold in the future across spatial scales. 2. Based on previous work on global invasion scenarios, we developed a workflow to downscale global scenarios to a regional and policy-relevant context. We applied this workflow at the European scale to create four European scenarios of biological invasions until 2050 that consider different environmental, socio-economic and socio-cultural trajectories, namely the European Alien Species Narratives (Eur-ASNs). 3. We compared the Eur-ASNs with their previously published global counterparts (Global-ASNs), assessing changes in 26 scenario variables. This assessment showed a high consistency between global and European scenarios in the logic and assumptions of the scenario variables. However, several discrepancies in scenario variable trends were detected that could be attributed to scale differences. This suggests that the workflow is able to capture scale-dependent differences across scenarios. 4. We also compared the Global- and Eur-ASNs with the widely used Global and European Shared Socioeconomic Pathways (SSPs), a set of scenarios developed in the context of climate change to capture different future socio-economic trends. Our comparison showed considerable divergences in the scenario space occupied by the different scenarios, with overall larger differences between the ASNs and SSPs than across scales (global vs. European) within the scenario initiatives. 5. Given the differences between the ASNs and SSPs, it seems that the SSPs do not adequately capture the scenario space relevant to understanding the complex future of biological invasions. This underlines the importance of developing independent but complementary scenarios focussed on biological invasions. The downscaling workflow we implemented and presented here provides a tool to develop such scenarios across different regions and contexts. This is a major step towards an improved understanding of all major drivers of global change, including biological invasions.