The gauge covariant magnetic perturbation theory is tailored for one-body Schrödinger operators perturbed by long-range magnetic fields. In this chapter, we present a self-contained exposition of the method, by outlining its technical foundations and discussing the physical heuristics behind the proofs. We apply it in order to prove the stability of spectral gaps and to study the location of the discrete spectrum. We also analyze the (lack of) continuity with respect to the magnetic field of spectral projections corresponding to finite spectral islands, which is a particularly important situation for systems modeling Chern insulators. Finally, we show how to construct approximate projections that have an explicit dependence with respect to the magnetic field parameter.