  We explore the intimate connection between spacetime geometry and
electrodynamics. This link is already implicit in the constitutive relations
between the field strengths and excitations, which are an essential part of the
axiomatic structure of electromagnetism, clearly formulated via integration
theory and differential forms. We briefly review the foundations of
electromagnetism based on charge and magnetic flux conservation, the Lorentz
force and the constitutive relations which introduce the spacetime metric. We
then proceed with the tensor formulation by assuming local, linear, homogeneous
and isotropic constitutive relations, and explore the physical, observable
consequences of Maxwell's equations in curved spacetime. The field equations,
charge conservation and the Lorentz force are explicitly expressed in general
(pseudo) Riemanian manifolds. The generalized Gauss and Maxwell-Amp\`{e}re
laws, as well as the wave equations, reveal potentially interesting
astrophysical applications. In all cases new electromagnetic couplings and
related phenomena are induced by spacetime curvature. The implications and
possible applications for gravity waves detection are briefly addressed. At the
foundational level, we discuss the possibility of generalizing the vacuum
constitutive relations, by relaxing the fixed conditions of homogeneity and
isotropy, and by assuming that the symmetry properties of the electro-vacuum
follow the spacetime isometries. The implications of this extension are briefly
discussed in the context of the intimate connection between electromagnetism
and the geometry (and causal structure) of spacetime.
