Structural systems are potentially subjected to damage initiating scenarios throughout the course of their service time. Depending on the nature and extent of the damaging event, they may experience significant reduction or even complete loss of their mechanical performance. This dissertation delves into the mechanics of structural systems under the notion of missing members from their domain, investigating types of structural systems: a) multi-story steel framed buildings, and b) materials with a truss-lattice microstructure. Part I of the dissertation investigates the performance of multi-story steel framed buildings under a column removal scenario, developing an analytical framework for their quasi-static robustness assessment. Two system-failure modes are taken into consideration: i) a yielding-type mechanism, where damage propagates as a series of beam plastic hinges or gravity connection failures, and ii) a stability-related mechanism, where a building column fails due to buckling. Validated against finite element results, this study demonstrates the capability of the method to assess the key features of the building performance such as the dominant collapse mode, the system capacity and the damage propagation path. Most importantly, it highlights that the governing mode is strongly dependent on the location of the initial damage scenario, emphasizing therefore the necessity for system-level approaches to identify correctly the building structural response. Part II of the dissertation draws attention to the emerging class of architected materials with a truss-lattice microstructure, and performs a detailed study on the impact of missing struts to their elastic mechanical behavior. Considering a list of truss-lattice topologies with varying coordination numbers (connectivity) and accounting for a series of defect scenarios, this work identifies the effect of the various topological parameters that govern the degradation rates of the elastic constants, such as the lattice connectivity, anisotropy, and interrelation of the pristine elastic constants. This investigation is supported by numerical (finite element modeling), experimental (two-photon lithography) and analytical (elastic micromechanical models) approaches. The results revealed that the behavior of periodic imperfect truss-lattices with coordination number Z≥12 is almost indistinguishable from homogeneous materials, and demonstrated a clear trajectory between the topology coordination number and the least deleterious defect arrangement.