Complex networks have been successfully used to describe the spreading of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize processes of social contagion, such as opinion formation or the adoption of novelties, where more complex dynamics of peer influence and reinforcement mechanisms are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and the contagion can occur, with different transmission rates, not only over the links but also through interactions in groups of different sizes. Numerical simulations of the model on both empirical data and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous with the formation of a bistable region where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes, and contribute to the understanding of the role of higher-order interactions in complex systems.