In vacuum the ``chiral neutrino oscillations'', i.e. the periodic transitions between the left- and right-handed states do not occur. The produced state differs from the chiral component that appear in the Lagrangian of interactions and should be computed for each specific process. The phase difference between components of a produced neutrino is space-time independent. This neutrino state consists of only positive energy solutions of the Dirac equation and therefore the energy splitting $2E_ν$ between the components with different helicities that would drive the chiral oscillations does not exist. Consideration in terms of neutrino propagators leads to the same conclusion. The situation is similar for the Majorana neutrinos and in the presence of flavor mixing. However, oscillations of the neutrino states produced in the chiral interactions are possible in matter with the length determined by the matter potential. Description of oscillations in terms of amplitudes of production and detection is elaborated that does not lead to any misconception. In the expanding Universe the relic neutrinos adiabatically convert to equal number densities of the left and right handed components.

22 pages,no figures. Prepared for special issue of Nucl.Phys.B "Clarifying common misconceptions in high energy physics and cosmology". Typos and references corrected; discussion in section 6 modified; version matches publication in Nucl. Phys.B