Our understanding of the dynamics of complex networked systems has increased significantly in the last two decades. However, most of our knowledge is built upon assuming pairwise relations among the system's components. This assumption is often an oversimplification, for instance, in social interactions that frequently occur within groups. To overcome this limitation, we use hypergraphs, which are mathematical objects that impose virtually no constraints regarding the sizes of the interactions. From a practical viewpoint, our initial motivation was social science's experimental studies on how committed minorities can overturn an established behavior. Despite that, our research questions generally focus on more fundamental aspects, such as the relationship between structure and dynamics when considering higher-order interactions and the constraints this kind of structure might impose. In this talk, we will discuss a straightforward generalization of contagion dynamics that account for a collection of groups modeled by critical-mass processes. We will show that our model has a rich and unexpected behavior beyond discontinuous transitions. In particular, we might have multistability and intermittency due to bimodal state distributions. We will also comment on other processes such as random walks, cascading failure, and failures and attacks to relate them to higher-order systems' structural aspects. We hope that our works can have applications advancing our understanding of real higher-order systems and that our theory might be helpful in applications such as hypergraph neural networks and convolutional networks on hypergraphs.