Complex networks are a powerful paradigm to model complex systems. Specific network models, e.g., multilayer networks, temporal networks, and signed networks, enrich the standard network representation with additional information to better capture real-world phenomena. Despite the keen interest in a variety of problems, algorithms, and analysis methods for these types of network, the problem of extracting cores and dense structures still has unexplored facets. In this work, we present advancements to the state of the art by the introduction of novel definitions and algorithms for the extraction of dense structures from complex networks, mainly cores. At first, we define core decomposition in multilayer networks together with a series of applications built on top of it, i.e., the extraction of maximal multilayer cores only, densest subgraph in multilayer networks, the speed-up of the extraction of frequent cross-graph quasi-cliques, and the generalization of community search to the multilayer setting. Then, we introduce the concept of core decomposition in temporal networks; also in this case, we are interested in the extraction of maximal temporal cores only. Finally, in the context of discovering polarization in large-scale online data, we study the problem of identifying polarized communities in signed networks. The proposed methodologies are evaluated on a large variety of real-world networks against naïve approaches, non-trivial baselines, and competing methods. In all cases, they show effectiveness, efficiency, and scalability. Moreover, we showcase the usefulness of our definitions in concrete applications and case studies, i.e., the temporal analysis of contact networks, and the identification of polarization in debate networks.