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The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, from human communications to chemical reactions and ecological systems, interactions can often occur in groups of three or more nodes and cannot be described simply in terms of dyads. Until recently little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can enhance our modeling capacities and help us understand and predict their dynamical behavior. Here we present a complete overview of the emerging field of networks beyond pairwise interactions. We discuss how to represent higher-order interactions and introduce the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review the measures designed to characterize the structure of these systems and the models proposed to generate synthetic structures, such as random and growing bipartite graphs, hypergraphs and simplicial complexes. We introduce the rapidly growing research on higher-order dynamical systems and dynamical topology, discussing the relations between higher-order interactions and collective behavior. We focus in particular on new emergent phenomena characterizing dynamical processes, such as diffusion, synchronization, spreading, social dynamics and games, when extended beyond pairwise interactions. We conclude with a summary of empirical applications, and an outlook on current modeling and conceptual frontiers.