New facilities to measure neutrino-nucleus cross sections, such as those possible in conjunction with spallation neutron sources, could provide an experimental foundation for the many neutrino-nucleus weak interaction rates needed in supernova models. This would enable more realistic supernova models and provide a greatly improved ability to understand the physics fundamental to supernovae by comparison of these models with detailed observations. Charged- and neutral-current neutrino interactions on nuclei in the stellar core play a central role in supernova dynamics and nucleosynthesis as well as being important for supernova neutrino detection. Measurements of these reactions on judiciously chosen targets would provide an invaluable test of the complex theoretical models used to compute the large number of neutrino-nucleus cross sections that are needed.