The difficulty of reconciling the staggering biodiversity found in tropical rainforests with classical theories of resource partitioning has led ecologists to explore neutral theories of coexistence, in which all species are assumed to have the same physiological parameters, and variations in species abundance arise from stochastic fluctuations. Simple neutral models have led to much progress, for example to the investigation of spatial and temporal ecological patterns and to the formulation of sampling models that allow contrasting theory and data. However, the high sensitivity of neutral models to slight perturbations of the parameters and the prediction of a strong correlation between a species’ abundance and its age are considered problematic. Here we propose a theory of coexistence in which all species have different physiological rates, and interact with each other through a network of competitive interactions. We show that our models produce robust coexistence of many species even when parameters are drawn at random. Importantly, the dynamical stability of our models is due to higher-order interactions — interactions involving more than two species at a time. The existence of higher-order interactions has been debated in ecology for decades, but their role in shaping ecological communities is still understudied. Our results show that higher-order interactions can have dramatic effects on the dynamics of ecological systems. When set in a stochastic framework, we recover many results from neutral theory but improve on the relationship between age and abundance.