In this talk, we discuss qualitative spectral properties of self-adjoint Dirac operators. We first briefly review some of the standard results for regular potentials and turn to more recent developments afterwards. Our main objective in this lecture is to discuss Dirac operators with singular potentials supported on curves or hyperplanes, where it is necessary to distinguish the so-called non-critical and critical cases for the strength of the singular perturbation. In particular, it turns out that Dirac operators with singular potentials in the critical case have some unexpected spectral properties that depend on the geometry of the support of the perturbation. This talk is based on joint some recent works with B. Benhellal, P. Exner, M. Holzmann, V. Lotoreichik, T. Ourmieres-Bonafos, and K. Pankrashkin.
