Abstract: The precise measurement of neutrino properties is among the highest priorities in fundamental particle physics, involving many experiments worldwide. Since the experiments rely on the interactions of neutrinos with bound nucleons inside atomic nuclei, the planned advances in the scope and precision of these experiments require a commensurate effort in the understanding and modeling of the hadronic and nuclear physics of these interactions, which is incorporated as a nuclear model in neutrino event generators. This model is essential to every phase of experimental analyses and its theoretical uncertainties play an important role in interpreting every result. In this White Paper we discuss in detail the impact of neutrino-nucleus interactions, especially the nuclear effects, on the measurement of neutrino properties using the determination of oscillation parameters as a central example. After an Executive Summary and a concise Overview of the issues, we explain how the neutrino event generators work, what can be learned from electron-nucleus interactions and how each underlying physics process – from quasi-elastic to deep inelastic scattering – is understood today. We then emphasize how our understanding must improve to meet the demands of future experiments. With every topic we find that the challenges can be met only with the active support and collaboration among specialists in strong interactions and electroweak physics that include theorists and experimentalists from both the nuclear and high energy physics communities.