In contrast to the assumption that social interactions only involve two individuals when modeling complex systems, larger interactions often occur in empirical settings. The collection of these group interactions form a higher-order interaction network, also known as a hypergraph. Dynamics on these higher-order networks can produce rich behavior, even for very simple models of contagion. We present two different contagion models: one representing social contagion and the other describing epidemiological contagion. For the social contagion model, we focus on the interplay between higher-order structure and the resulting contagion dynamics of a hypergraph SIS model; in particular, we describe how degree heterogeneity and community structure can affect the onset and existence of tipping-point behavior and polarization. For the epidemiological model, we describe a reality-inspired model of environmentally-mediated contagion and present a method to infer network structure and epidemiological parameters from time-series data. We discuss initial work and challenges applying this model to transmission of C. diff in a hospital setting.