Rainforests, coral reefs and other very large ecosystems seem to be the most stable in nature, but this has long been regarded as mathematically paradoxical. More generally, the relationship between structure and dynamics in complex systems is the subject of much debate. I will discuss how 'trophic coherence', a recently identified property of food webs and other biological networks, is key to understanding many dynamical and structural features of complex systems. In particular, it allows networks to become more stable with increasing size and complexity, determines whether a given system will be in a regime of high or of negligible feedback, and influences spreading processes such as epidemics or cascades of neural activity.