Biological invasions are a global consequence of an increasingly connected world and the rise in human population size. The numbers of invasive alien species – the subset of alien species that spread widely in areas where they are not native, affecting the environment or human livelihoods – are increasing. Synergies with other global changes are exacerbating current invasions and facilitating new ones, thereby escalating the extent and impacts of invaders. Invasions have complex and often immense long‐term direct and indirect impacts. In many cases, such impacts become apparent or problematic only when invaders are well established and have large ranges. Invasive alien species break down biogeographic realms, affect native species richness and abundance, increase the risk of native species extinction, affect the genetic composition of native populations, change native animal behaviour, alter phylogenetic diversity across communities, and modify trophic networks. Many invasive alien species also change ecosystem functioning and the delivery of ecosystem services by altering nutrient and contaminant cycling, hydrology, habitat structure, and disturbance regimes. These biodiversity and ecosystem impacts are accelerating and will increase further in the future. Scientific evidence has identified policy strategies to reduce future invasions, but these strategies are often insufficiently implemented. For some nations, notably Australia and New Zealand, biosecurity has become a national priority. There have been long‐term successes, such as eradication of rats and cats on increasingly large islands and biological control of weeds across continental areas. However, in many countries, invasions receive little attention. Improved international cooperation is crucial to reduce the impacts of invasive alien species on biodiversity, ecosystem services, and human livelihoods. Countries can strengthen their biosecurity regulations to implement and enforce more effective management strategies that should also address other global changes that interact with invasions.

Biological invasions have steadily increased over recent centuries. However, we still lack a clear expectation about future trends in alien species numbers. In particular, we do not know whether alien species will continue to accumulate in regional floras and faunas, or whether the pace of accumulation will decrease due to the depletion of native source pools. Here, we apply a new model to simulate future numbers of alien species based on estimated sizes of source pools and dynamics of historical invasions, assuming a continuation of processes in the future as observed in the past (a business‐as‐usual scenario). We first validated performance of different model versions by conducting a back‐casting approach, therefore fitting the model to alien species numbers until 1950 and validating predictions on trends from 1950 to 2005. In a second step, we selected the best performing model that provided the most robust predictions to project trajectories of alien species numbers until 2050. Altogether, this resulted in 3,790 stochastic simulation runs for 38 taxon–continent combinations. We provide the first quantitative projections of future trajectories of alien species numbers for seven major taxonomic groups in eight continents, accounting for variation in sampling intensity and uncertainty in projections. Overall, established alien species numbers per continent were predicted to increase from 2005 to 2050 by 36%. Particularly, strong increases were projected for Europe in absolute (+2,543 ± 237 alien species) and relative terms, followed by Temperate Asia (+1,597 ± 197), Northern America (1,484 ± 74) and Southern America (1,391 ± 258). Among individual taxonomic groups, especially strong increases were projected for invertebrates globally. Declining (but still positive) rates were projected only for Australasia. Our projections provide a first baseline for the assessment of future developments of biological invasions, which will help to inform policies to contain the spread of alien species.

Summary: Understanding how invasions by exotic plant species occur has been and still is a fundamental goal in the field of invasion ecology. Recently, research focus has shifted to below‐ground mechanisms of invasion, which has provided valuable insights into the role of soil microbes in the invasion process. Research on how soil microbes drive plant invasions has produced a multitude of potential mechanisms and hypotheses, and we have taken this timely opportunity to organize these hypotheses, explore interrelations among them and with other drivers of invasion and identify areas requiring more in‐depth study. Synthesis. This special feature brings together current research on soil microbes as drivers and responders of invasion, highlighting the methods that can be used to advance our understanding of how, when and which soil microbes play a role in the invasion process. Further advances in the field can be made by increasing realism in experiments, making wider use of molecular methods and culturing of microbial isolates and considering the importance of microbes relative to other invasion drivers.