The level of resources required for scientific research and the extent of public support increased spectacularly since WWII. It was fueled by the expectation that providing more resources would support economic growth, create jobs, enhance social welfare, protect the environment, and expand the frontiers of human knowledge. The diversity of goals also turned into an increasing complexity and differentiation of the public funding landscape, in terms of the underlying motivations, of funding instruments and of the R&D performers receiving public funds. This introductory chapter to the Handbook of Public Research Funding reviews the state of the art on these topics and summarizes the main insights of the individual chapters. To deal with vertical (across system’s layers) and horizontal (between different narratives, instruments and beneficiaries) complexity, it is suggested to draw from theoretical lenses from fields such as sociology of markets, public policy, management and strategies in order to develop analytical approaches that allow analyzing public research funding in a more differentiated manner. While until now the multiplication of funding instruments has been the prevalent approach to address new policy issues in public research funding, it is argued that this expansion has come to limits and that managing complexity and dealing with complementarities and interactions between existing instruments, agencies and performers will become increasingly important in the future.