  We assume that the Pauli exclusion principle is violated for neutrinos, and
consequently, neutrinos obey the Bose-Einstein statistics. Cosmological and
astrophysical consequences of this assumption are considered. Neutrinos may
form cosmological Bose condensate which accounts for all (or a part of) the
dark matter in the universe. ``Wrong'' statistics of neutrinos could modify big
bang nucleosynthesis, leading to the effective number of neutrino species
smaller than three. Dynamics of the supernova collapse would be influenced and
spectra of the supernova neutrinos may change. The presence of neutrino
condensate would enhance contributions of the Z-bursts to the flux of the UHE
cosmic rays and lead to substantial refraction effects for neutrinos from
remote sources. The Pauli principle violation for neutrinos can be tested in
the two-neutrino double beta decay.
