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. 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. Moving from deterministic to stochastic models, we find that the presence of higher-order interactions, which make equilibrium points attractive, dramatically increases the time to extinction in isolated systems, allowing for the prolonged coexistence of species. When we let the system evolve, we recover many empirically observed macroecological patterns.