In this short review we discuss the notion of lepton numbers. The strong evidence in favor of neutrino oscillations obtained recently in the super-Kamiokande atmospheric neutrino experiment and in solar neutrino experiments imply that the law of conservation of family lepton numbers Le, Lμ and Lτ is strongly violated. We consider the states of flavor neutrinos νe, νμ and ντ and we discuss the evolution of these states in space and time in the case of nonconservation of family lepton numbers due to the mixing of light neutrinos. We discuss and compare different flavor neutrino discovery experiments. We stress that experiments on the search for νμ-->ντ and νe-->ντ oscillations demonstrated that the flavor neutrino ντ is a new type of neutrino, different from νe and νμ. In the case of neutrino mixing, the lepton number (only one) is connected with the nature of massive neutrinos. Such a conserved lepton number exists if massive neutrinos are Dirac particles. We review possibilities to check in future experiments whether the conserved lepton number exists.