Alien species are a major component of human-induced environmental change. Variation in the numbers of alien species found in different areas is likely to depend on a combination of anthropogenic and environmental factors, with anthropogenic factors affecting the number of species introduced to new locations, and when, and environmental factors influencing how many species are able to persist there. However, global spatial and temporal variation in the drivers of alien introduction and species richness remain poorly understood. Here, we analyse an extensive new database of alien birds to explore what determines the global distribution of alien species richness for an entire taxonomic class. We demonstrate that the locations of origin and introduction of alien birds, and their identities, were initially driven largely by European (mainly British) colonialism. However, recent introductions are a wider phenomenon, involving more species and countries, and driven in part by increasing economic activity. We find that, globally, alien bird species richness is currently highest at midlatitudes and is strongly determined by anthropogenic effects, most notably the number of species introduced (i.e., “colonisation pressure”). Nevertheless, environmental drivers are also important, with native and alien species richness being strongly and consistently positively associated. Our results demonstrate that colonisation pressure is key to understanding alien species richness, show that areas of high native species richness are not resistant to colonisation by alien species at the global scale, and emphasise the likely ongoing threats to global environments from introductions of species.