  Within the scope of Bianchi type-$I$ cosmological model we have studied the
role of spinor field in the evolution of the Universe. In doing so we have
considered the case with non-minimal coupling. It was found that the
non-diagonal components of the energy-momentum tensor of the spinor field,
hence the restrictions on the space-time geometry remain the same as in case of
minimal coupling. Since in this case the diagonal components of the
energy-momentum tensor differ, the evolution of the corresponding universe also
differs. For example, while a linear spinor field with non-minimal coupling or
nonlinear spinor field with minimal coupling give rise to open universe, a
nonlinear spinor field with non-minimal coupling with the same parameters can
generate close universe that at the beginning expands, and after attaining some
maximum value begin to contract and finally ends in a Big Crunch.
