With the discovery of neutrino oscillations and the resulting measurement of lepton flavor mixing, new and exciting facets of the flavor puzzle of the Standard Model have been revealed. In this talk, two stories will be told about the possible origins of ultralight neutrino masses and large lepton flavor mixing angles. The first is about the possibility of naturally suppressed Dirac neutrino masses in models with additional low energy gauge symmetries. The second focuses on using icosahedral (A5) flavor symmetry to implement the intriguing idea that the solar neutrino mixing angle is related to the famed "golden ratio."