The basic interaction unit of many dynamical systems involves more than two nodes. In such situations where networks are not an appropriate modelling framework, it has recently become increasingly popular to turn to higher-order models, including hypergraphs. Multi-body interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. In this collection of papers, we derive and analyse models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. These models reveal that multi-body dynamical effects that go beyond rescaled pairwise interactions can only appear if the interaction function is non-linear, regardless of the underlying multi-body structure. As a practical application, building on sociological theory, we introduce a a non-linear function which can model different from sociological theory motivated phenomena such as peer pressure and stubbornness. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state.The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure and the form of the interaction function.