Doctor of Philosophy / Department of Electrical and Computer Engineering / Caterina Scoglio / Critical infrastructures are repeatedly attacked by external triggers causing tremendous amount of damages. Any infrastructure can be studied using the powerful theory of complex networks. A complex network is composed of extremely large number of different elements that exchange commodities providing significant services. The main functions of complex networks can be damaged by different types of attacks and failures that degrade the network performance. These attacks and failures are considered as disturbing dynamics, such as the spread of viruses in computer networks, the spread of epidemics in social networks, and the cascading failures in power grids. Depending on the network structure and the attack strength, every network differently suffers damages and performance degradation. Hence, quantifying the robustness of complex networks becomes an essential task. In this dissertation, new metrics are introduced to measure the robustness of technological and social networks with respect to the spread of epidemics, and the robustness of power grids with respect to cascading failures. First, we introduce a new metric called the Viral Conductance ($VC_{SIS}$) to assess the robustness of networks with respect to the spread of epidemics that are modeled through the susceptible/infected/susceptible ($SIS$) epidemic approach. In contrast to assessing the robustness of networks based on a classical metric, the epidemic threshold, the new metric integrates the fraction of infected nodes at steady state for all possible effective infection strengths. Through examples, $VC_{SIS}$ provides more insights about the robustness of networks than the epidemic threshold. In addition, both the paradoxical robustness of Barab\'si-Albert preferential attachment networks and the effect of the topology on the steady state infection are studied, to show the importance of quantifying the robustness of networks. Second, a new metric $VC_$ is introduced to assess the robustness of networks with respect to the spread of susceptible/infected/recovered ($SIR$) epidemics. To compute $VC_$, we propose a novel individual-based approach to model the spread of $SIR$ epidemics in networks, which captures the infection size for a given effective infection rate. Thus, $VC_$ quantitatively integrates the infection strength with the corresponding infection size. To optimize the $VC_$ metric, a new mitigation strategy is proposed, based on a temporary reduction of contacts in social networks. The social contact network is modeled as a weighted graph that describes the frequency of contacts among the individuals. Thus, we consider the spread of an epidemic as a dynamical system, and the total number of infection cases as the state of the system, while the weight reduction in the social network is the controller variable leading to slow/reduce the spread of epidemics. Using optimal control theory, the obtained solution represents an optimal adaptive weighted network defined over a finite time interval. Moreover, given the high complexity of the optimization problem, we propose two heuristics to find the near optimal solutions by reducing the contacts among the individuals in a decentralized way. Finally, the cascading failures that can take place in power grids and have recently caused several blackouts are studied. We propose a new metric to assess the robustness of the power grid with respect to the cascading failures. The power grid topology is modeled as a network, which consists of nodes and links representing power substations and transmission lines, respectively. We also propose an optimal islanding strategy to protect the power grid when a cascading failure event takes place in the grid. The robustness metrics are numerically evaluated using real and synthetic networks to quantify their robustness with respect to disturbing dynamics. We show that the proposed metrics outperform the classical metrics in quantifying the robustness of networks and the efficiency of the mitigation strategies. In summary, our work advances the network science field in assessing the robustness of complex networks with respect to various disturbing dynamics.