One of the most powerful properties of mathematical systems theory is the fact that interconnecting systems yields composites that are themselves systems. This property allows for the engineering of complex systems by aggregating simpler systems into intricate patterns. We call these interconnection patterns the "structure" of the system. Similarly, this property also enables the understanding of complex systems by decomposing them into simpler parts. We likewise call the relationship between these parts the "structure" of the system. At first glance, these may appear to represent identical views of structure of a system. However, further investigation invites the question: are these two notions of structure of a system the same? This dissertation answers this question by developing a theory of dynamical structure. The work begins be distinguishing notions of structure from their associated mathematical representations, or models, of a system. Focusing on linear time invariant (LTI) systems, the key technical contributions begin by extending the definition of the dynamical structure function to all LTI systems and proving essential invariance properties as well as extending necessary and sufficient conditions for the reconstruction of the dynamical structure function from data. Given these extensions, we then develop a framework for analyzing the structures associated with different representations of the same system and use this framework to show that interconnection (or subsystem) structures are not necessarily the same as decomposition (or signal) structures. We also show necessary and sufficient conditions for the reconstruction of the interconnection (or subsystem) structure for a class of systems. In addition to theoretical contributions, this work also makes key contributions to specific applications. In particular, network reconstruction algorithms are developed that extend the applicability of existing methods to general LTI systems while improving the computational complexity. Also, a passive reconstruction method was developed that enables reconstruction without actively probing the system. Finally, the structural theory developed here is used to analyze the vulnerability of a system to simultaneous attacks (coordinated or uncoordinated), enabling a novel approach to the security of cyber-physical-human systems.