Invasive species are an excellent opportunity to think about the nature society desires, particularly in the face of global changes. Nature and human views of nature are rapidly evolving; our approach to biological invasions through biosecurity institutions and land management policies must evolve in tandem with these changes. We review three dimensions that are insufficiently addressed. First, biological invasions are culturally shaped and interpreted. Humans play a major role in the movement and nurturing of alien life, and esthetics, perception, and emotion are deeply implicated in the management of invasive species. What people fear or regret with invasive species are not their effects on nature per se, but their effects on a particular desired nature, and policymaking must reflect this. Second, biological invasions are not restricted to negative impacts. Invasions take place in landscapes where many natural conditions have been altered, so policy tools must recognize that invasive species are a functional, structural, and compositional part of transformed ecosystems. In some cases, native species benefit from changes in resource availability caused by invasions or from protections provided by an invasive plant. Finally, invasive species can help ecosystems and people to adapt to global change by maintaining ecosystem processes such as productivity, carbon storage, and nutrient cycling in a context of climate change or land cover transformations. While recognition is growing among ecologists that novel, invaded ecosystems have value, and while the on-the-ground application of biosecurity policies has of necessity adjusted to local contexts and other agendas, invasion biology could aid policymaking by better addressing the three complexities inherent in the three dimensions highlighted above.