  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 L_e,
L_mu and L_tau is strongly violated. We consider the states of flavor neutrinos
nu_e, nu_mu and nu_tau and we discuss the evolution of these states in space
and time in the case of non-conservation 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
nu_mu->nu_tau and nu_e->nu_tau oscillations demonstrated that the flavor
neutrino nu_tau is a new type of neutrino, different from nu_e and nu_mu. In
the case of neutrino mixing, the lepton number (only one) is connected with the
nature of massive neutrinos. Such conserved lepton number exist if massive
neutrinos are Dirac particles. We review possibilities to check in future
experiments whether the conserved lepton number exists.
