Tailored graph ensembles are a developing bridge between statistical mechanics and biological networks. In this thesis, this concept is used to generate a suite of rigorous mathematical tools to quantify and compare the topology of cellular signalling networks. Earlier published results to quantify the entropy of constrained random graph ensembles are extended by looking at constraints relating to directed graphs, bipartite graphs, neighbourhood compositions and generalised degrees. To incorporate constraints relating to the average number of short loops, a number of innovative techniques are reviewed and extended, moving the analysis beyond the usual tree-like assumption. The generation of unbiased sample networks under some of these new constraints is studied. A series of illustrations of how these concepts may be applied to systems biology are developed. Topological observables are obtained from real biological networks and the entropy of the associated random graph ensemble is calculated. Certain studies on over-represented motifs are replicated and the influence of the choice of null model is considered. The correlation between the topological role of each protein and its lethality is studied in yeast. Throughout, this document aims to promote looking at a network as a realisation satisfying certain constraints rather than just as a list of nodes and edges. This may initially seem to be an abstract approach, but it is in fact a more natural viewpoint within which to consider many fundamental questions regarding the origin, function and design of observed real networks.