In the course of the growth of the Internet and due to increasing availability of data, over the last two decades, the field of network science has established itself as an own area of research. With quantitative scientists from computer science, mathematics, and physics working on datasets from biology, economics, sociology, political sciences, and many others, network science serves as a paradigm for interdisciplinary research. One of the major goals in network science is to unravel the relationship between topological graph structure and a network's function. As evidence suggests, systems from the same fields, i.e. with similar function, tend to exhibit similar structure. However, it is still vague whether a similar graph structure automatically implies likewise function. This dissertation aims at helping to bridge this gap, while particularly focusing on the role of triadic structures. After a general introduction to the main concepts of network science, existing work devoted to the relevance of triadic substructures is reviewed. A major challenge in modeling such structure is the fact that not all three-node subgraphs can be specified independently of each other, as pairs of nodes may participate in multiple triadic subgraphs. In order to overcome this obstacle, a novel class of generative network models based on pair-disjoint triadic building blocks is suggested. It is further investigated whether triad motifs - subgraph patterns which appear significantly more frequently than expected at random - occur homogeneously or heterogeneously distributed over graphs. Finally, the influence of triadic substructure on the evolution of dynamical processes acting on their nodes is studied. It is observed that certain motifs impose clear signatures on the systems' dynamics, even when embedded in a larger network structure.