Abstract : Quantum coherence can significantly increase the strength of the forward scattering of neutrinos propagating through the Earth and interacting with matter. The index of refraction of the neutrinos propagating in a medium and hence their phase velocity is determined by the forward scattering. So, depending on the nature of the interaction of neutrinos with matter, their phase velocity can be larger than the speed of light in vacuum. We show that such effects can explain the apparent superluminal propagation of muon neutrinos found recently by the OPERA experiment. Our proposal explains why the neutrino oscillations and the propagation of neutrinos from supernova 1987A are unaffected. It can be verified by changing the amount of neutrino coherence or by changing the composition of matter in which they propagate.