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Recent studies suggest that the emergence of cooperative behavior can be explained by generalized reciprocity, a behavioral mechanism based on the principle of "help anyone if helped by someone". In complex systems, the cooperative dynamics is largely determined by the network structure which dictates the interactions among neighboring individuals. Despite an abundance of studies, the role of the network structure in in promoting cooperation through generalized reciprocity remains an under-explored phenomenon. In this doctoral thesis, we utilize basic tools from the dynamical systems theory, and develop a unifying framework for investigating the cooperation dynamics of generalized reciprocity on complex networks. We use this framework to present a theoretical overview on the role of generalized reciprocity in promoting cooperation in three distinct interaction structures: i) social dilemmas, ii) multidimensional networks, and iii) fluctuating environments. The results suggest that cooperation through generalized reciprocity always emerges as the unique attractor in which the overall level of cooperation is maximized, while simultaneously exploitation of the participating individuals is prevented. The effect of the network structure is captured by a local centrality measure which uniquely quantifies the propensity of the network structure to cooperation, by dictating the degree of cooperation displayed both at microscopic and macroscopic level. As a consequence, the implementation of our results may go beyond explaining the evolution of cooperation. In particular, they can be directly applied in domains that deal with the development of artificial systems able to adequately mimic reality, such as reinforcement learning.