Is the Zee model the model of neutrino masses?

The Zee model predicts naturally two heavy, strongly degenerate and almost maximally mixed neutrinos and one light neutrino with small mixing. This pattern coincides with the one needed for a solution of the atmospheric neutrino problem by ${\nu }_{\mu }\to {\nu }_{\tau }$ oscillations and for existence of the two component hot dark matter in the Universe. Furthermore, the oscillations ${\overline{\nu }}_{\mu }\to {\overline{\nu }}_e$ can be in the range of sensitivity of KARMEN, LSND experiments. The phenomenology of this scenario is considered and the possibility to check it in the forthcoming experiments is discussed. The scenario implies large values and inverse flavor hierarchy of couplings of the Zee boson with fermions: $f_{e\tau }\ll f_{\mu \tau }<~f_{e\mu }\sim 0.1$. The main signatures of the scenario are a strongly suppressed signal of ${\nu }_{\mu }\to {\nu }_{\tau }$ oscillation in CHORUS and NOMAD experiments, so that a positive result from these experiments will rule out the scenario, the possibility of the observation of ${\nu }_e\to {\nu }_{\tau }$ oscillations by CHORUS and NOMAD, the corrections to the muon decay and neutrino-electron scattering at the level of experimental errors, and a branching ratio $B(\mu \to e\gamma )$ bigger than ${10}^{-13}$. The solar neutrino problem can be solved by the introduction of an additional very light singlet fermion without appreciable modification of the active neutrino pattern.