A standard problem in complex systems science has been to understand how infectious diseases, information, or any other contagion can spread within a system. Simple models of contagions tend to assume random mixing of elements, but real interactions are not random pairwise encounters: they occur within clearly defined higher-order structures. These higher-level structures could represent communities in social systems, cells in organisms or modules in neural networks. For a broader understanding of contagion dynamics in complex networks, we need to embrace higher-order structure, which can itself take many forms such as simplicial complexes or hypergraphs. To accurately describe spreading processes on these higher-order networks and correctly account for the heterogeneity of the underlying structure, we use a set of approximate master equations. This general framework allows us to unveil and characterize important properties of these systems. Here we focus on three of them: the localization of contagions within certain substructures, the bistability of the stationary state and a crossover of the optimal seeding strategies to maximize early spread.