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Communities that are very rich in species could persist thanks to the stabilizing role of higher-order interactions, in which the presence of a species influences the interaction between other species.Ecologists have long sought a way to explain how the remarkable biodiversity observed in nature is maintained. On the one hand, simple models of interacting competitors cannot produce the stable persistence of very large ecological communities1,2,3,4,5. On the other hand, neutral models6,7,8,9, in which species do not interact and diversity is maintained by immigration and speciation, yield unrealistically small fluctuations in population abundance10, and a strong positive correlation between a species' abundance and its age11, contrary to empirical evidence. Models allowing for the robust persistence of large communities of interacting competitors are lacking. Here we show that very diverse communities could persist thanks to the stabilizing role of higher-order interactions12,13, in which the presence of a species influences the interaction between other species. Although higher-order interactions have been studied for decades14,15,16, their role in shaping ecological communities is still unclear5. The inclusion of higher-order interactions in competitive network models stabilizes dynamics, making species coexistence robust to the perturbation of both population abundance and parameter values. We show that higher-order interactions have strong effects in models of closed ecological communities, as well as of open communities in which new species are constantly introduced. In our framework, higher-order interactions are completely defined by pairwise interactions, facilitating empirical parameterization and validation of our models.