We apply the Landau damping theory to collective flavor excitations in dense neutrino media. We demonstrate explicitly that the (meta)stable modes of the collective flavor excitations in a physical dense neutrino gas are indeed weakly damped. We show that, in a neutrino medium with a continuous energy distribution, there can exist infinitely many dispersion relation branches of collective flavor excitations that extend to large wave numbers. In a neutrino gas with one or more zero crossings in the distribution of the difference of the (neutrino) lepton numbers (DLNs), the least damped resonant branches and the metastable branches can suffer from inverse Landau damping at certain wave numbers and thus become unstable. We also introduce the concept of convolved DLN distributions, which can be used to understand the physical origin of the slow and fast flavor instabilities in dense neutrino media in core-collapse supernovae and neutron star mergers.