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In this paper, we provide an overview of the research conducted in the context of structural systems since the latest survey by Dion et al. in 2003. We systematically consider all the papers that cite this survey as well as the seminal work in this field that took place on and after the publication of the later survey, are published in peer-reviewed venues and in English. Structural systems theory deals with parametric systems where parameters might be unknown and, therefore, addresses the study of systems properties that depend only on the system's structure (or topology) described by the inter-dependencies between state variables. Remarkably, structural systems properties hold generically (i.e., almost always) under the assumption that parameters are independent. Therefore, it constitutes an approach to assess necessary conditions that systems should satisfy. In recent years, structural systems theory was applied to design systems that attain such properties, as well as to endure resilient/security and privacy properties. Furthermore, structural systems theory enables the formulation of such topics as combinatorial optimization problems, which allow us to understand their computational complexity and find algorithms that are efficiently deployed in the context of large-scale systems. In particular, we present an overview of how structural systems theory has been used in the context of linear time-invariant systems, as well as other dynamical models, for which a brief description of the different problem statements and solutions approaches are presented. Next, we describe recent variants of structural systems theory, as well as different applications of the classical and new approaches. Finally, we provide an overview of recent and future directions in this field.