Invasive alien species have become persistent features of landscapes, ecological communities and ecosystems across much of the world. In most regions the problem is expected to grow as new species continue to establish and spread. Their effects on communities are twofold - they both enrich local species richness and shift patterns of dominance. While the number of globally established alien taxa is large, the subset with evidence of negative biodiversity impact, i.e. invasive alien species (IAS), is considerably smaller, and recent work suggests that in some regions these numbers are beginning to stabilise. In the Antarctic, for example, although around 600 alien species have been introduced, mostly since 1950, recent rates of introduction are either slowing or stable. Nonetheless, local range expansion of already-established IAS continues worldwide, and their impact on biodiversity grows as a function of the area they occupy. The richness and dominance effects of alien species on local communities are distinct but can interact, together transforming community properties and function. Managing these dynamics is complicated by two challenges. First, population eradication or control is expensive and often fails. Second, under climate change it may become increasingly difficult to distinguish deliberate human introductions from natural range expansion, complicating decisions about appropriate management responses. Looking ahead, how should novel ecosystems be approached, i.e. those characterised by mixed native and alien species communities? I argue that two key tasks now face invasion biologists and ecologists: (i) applying our substantial understanding of biological invasions to guide and advocate for future-ready governance, and (ii) developing model-based forecasts of likely futures in the presence of IAS. Both tasks point toward the need to think beyond prevention and control, toward envisioning the desired qualities of natural systems and actively managing to maintain those values. The resist-adapt-direct framework and integrated environmental governance offer structured ways of approaching these policy and management challenges, while modelled trajectories of community change can inform expectations. Importantly, the rapidly growing and openly accessible knowledge base on IAS makes it increasingly feasible to forecast what ecological communities might look like fifty to a hundred years from now. Such forecasts can inform and prepare society, and support the formulation of policy that is ready for the ecological futures we are likely to face.