Many of the global challenges that confront humanity are interlinked in a dynamic complex network, with multiple feedback loops, nonlinear interactions and interdependencies that make it difficult, if not impossible, to consider individual threats in isolation. These challenges are mainly dealt with, however, by considering individual threats in isolation (at least in political terms). The mitigation of dual climate and biodiversity threats, for example, is linked to a univariate 1.5°C global warming boundary and a global area conservation target of 30% by 2030. The situation has been somewhat improved by efforts to account for interactions through multidimensional target setting, adaptive and open management and market-based decision pathways. But the fundamental problem still remains—that complex systems such as those formed by the network of global threats have emergent properties that are more than the sum of their parts. We must learn how to deal with or live with these properties if we are to find effective ways to cope with the threats, individually and collectively. Here, we argue that recent progresses in complex systems research and related fields have enhanced our ability to analyse and model such entwined systems to the extent that it offers the promise of a new approach to sustainability. We discuss how this may be achieved, both in theory and in practice, and how human cultural factors play an important but neglected role that could prove vital to achieving success. Read the free Plain Language Summary for this article on the Journal blog.