  After a brief review of the foundations of (pre-metric) electromagnetism, we
explore some physical consequences of electrodynamics in curved spacetime. In
general, new electromagnetic couplings and related phenomena are induced by the
spacetime curvature. The applications of astrophysical interest considered here
correspond essentially to the following geometries: the Schwarzschild spacetime
and the spacetime around a rotating spherical mass in the weak field and slow
rotation regime. In the latter, we use the Parameterised Post-Newtonian (PPN)
formalism. We also explore the hypothesis that the electric and magnetic
properties of vacuum reflect the spacetime isometries. Therefore, the
permittivity and permeability tensors should not be considered homogeneous and
isotropic a priori. For spherical geometries we consider the effect of relaxing
the homogeneity assumption in the constitutive relations between the fields and
excitations. This affects the generalized Gauss and Maxwell-Amp\`{e}re laws
where the electric permittivity and magnetic permeability in vacuum depend on
the radial coordinate in accordance with the local isometries of space. For the
axially symmetric geometries we relax both the assumptions of homogeneity and
isotropy. We explore simple solutions and discuss the physical implications
related to different phenomena such as: the decay of electromagnetic fields in
the presence of gravity, magnetic terms in Gauss law due to the
gravitomagnetism of the spacetime around rotating objects, a frame-dragging
effect on electric fields and the possibility of a spatial (radial) variability
of the velocity of light in vacuum around spherical astrophysical objects for
strong gravitational fields.
