This thesis focuses on the topological characterization of complex networks. It specifically focuses on those elementary graph measures that are of interest when quantifying topology-related aspects of the robustness of complex networks. This thesis makes the following contributions to the field of complex networks. Firstly, the thesis analyses the relationships among a variety of graph measures and proposes a definite set, capable of expressing the most relevant topological properties of complex networks. Secondly, the thesis relies on spectral measures to initiate a classification of the qualitative topological properties that characterize specific classes of complex networks. Thirdly, the thesis illustrates the use of spectral measures in a quantitative characterization of topology-related aspects of the network robustness. Finally, this thesis demonstrates the use of spectral measures in a quantitative characterization of the extent to which the robustness to different types of failures manifests itself in the underlying complex networks structure.