We consider oscillations of neutrinos under conditions in which the neutrino density is sufficiently large that neutrino-neutrino interactions cannot be neglected. A formalism is developed to treat this highly nonlinear system. Numerical analysis reveals a rich array of phenomena. In certain gases, a self-induced Mikheyev-Smirnov-Wolfenstein effect occurs in which electron neutrinos are resonantly converted into muon neutrinos. In another relatively low-density gas, an unexpected parametric resonant conversion takes place. Finally, neutrino-neutrino interactions maintain coherence in one system for which a priori one expected decoherence.