  We investigate the effect of non-symmetric relatively bounded perturbations
on the spectrum of self-adjoint operators. In particular, we establish
stability theorems for one or infinitely many spectral gaps along with
corresponding resolvent estimates. These results extend, and improve, classical
perturbation results by Kato and by Gohberg/Krein. Further, we study essential
spectral gaps and perturbations exhibiting additional structure with respect to
the unperturbed operator; in the latter case, we can even allow for
perturbations with relative bound $\ge 1$. The generality of our results is
illustrated by several applications, massive and massless Dirac operators,
point-coupled periodic systems, and two-channel Hamiltonians with dissipation.
