The discovery of neutrino masses and flavor mixing represents a breakthrough in the search for physics beyond the Standard Model. As the field of neutrino physics enters the precision era, neutrinos are ubiquitous and yet retain many mysteries, key to particle physics, nuclear physics, and astrophysics. At present, we are faced with a number of urgent questions, the resolution of which will require multidisciplinary collaborations. For example: 1. Considerable advances have occurred in hadronic and nuclear physics over the last decade. In particular, lattice QCD simulations of nucleon form factors with physical quark masses are now available. To what extent do all these developments improve the accuracy with which neutrino oscillation information will be extracted in the future? 2. Several recent new-physics scenarios predict the existence of new neutrino states, with hidden (dark sector) interactions. Will we have a sufficiently accurate understanding of experimental data to stress-test the three-flavor oscillation paradigm and examine signatures of new physics? 3. New levels of precision will be reached in the next few years both in short-baseline experiments and in cosmological surveys. What is the interplay between these two datasets? 4. Supernova simulations have entered a new era with three-dimensional simulations. Can the neutrino signal from the next galactic supernova unravel the dynamics of the explosion mechanism? Can this signal be correlated with the predicted nucleosynthetic yields? 5. What are the sources of the ultra-high-energy neutrinos detected at IceCube?