This chapter presents recent advances in the field of structural robustness and progressive collapse of deteriorating structural systems, with emphasis on the relationships among structural robustness, static indeterminacy, structural redundancy, and failure times. Damage is viewed as a progressive deterioration of the material properties and its amount is evaluated at the member level by means of a damage index associated with prescribed patterns of cross-sectional deterioration. The variation of suitable performance indicators compared with the amount of damage is used to formulate dimensionless measures of structural robustness. An index of structural integrity is defined to quantify the severity of the structural failure with respect to its consequences. The role of damage propagation on structural robustness is investigated by considering different propagation mechanisms and by using a damage-sensitive fault-tree analysis. The role of structural robustness on progressive collapse, as well as the relationship between structural robustness and static indeterminacy, are also investigated by considering parallel and mixed series-parallel truss deteriorating systems with various degrees of static indeterminacy. Time-variant measures of structural robustness and redundancy are developed with respect to the loads associated to the first local failure and to the structural collapse. The elapsed time between these two types of failures is investigated as a measure of the ability of the system to be repaired after local failure. This approach is illustrated through the application to a reinforced concrete frame under different corrosion damage scenarios.