The growing availability of multirelational data gives rise to an opportunity for novel characterization of complex real-world relations, supporting the proliferation of diverse network models such as Attributed Graphs, Heterogeneous Networks, Multilayer Networks, Temporal Networks, Location-aware Networks, Knowledge Networks, Probabilistic Networks, and many other task-driven and data-driven models. In this paper, we propose an overview of these models and their main applications, described under the common denomination of Feature-rich Networks, i. e. models where the expressive power of the network topology is enhanced by exposing one or more peculiar features. The aim is also to sketch a scenario that can inspire the design of novel feature-rich network models, which in turn can support innovative methods able to exploit the full potential of mining complex network structures in domain-specific applications.