  We study the zeta-regularized determinant of a non self-adjoint elliptic
operator on a closed odd-dimensional manifold. We show that, if the spectrum of
the operator is symmetric with respect to the imaginary axis, then the
determinant is real and its sign is determined by the parity of the number of
the eigenvalues of the operator, which lie on the positive part of the
imaginary axis. It follows that, for many geometrically defined operators, the
phase of the determinant is a topological invariant. In numerous examples,
coming from geometry and physics, we calculate the phase of the determinants in
purely topological terms. Some of those examples were known in physical
literature, but no mathematically rigorous proofs and no general theory were
available until now.
