I present various regularity results for the time-dependent Maxwell's equations. The general results, based on the standard semi-group and variational theories, are valid for any Lipschitz domain. They are optimal on the scales C^\alpha (in time) and H^s (in space) for smooth or convex domains.

This is no longer the case for domains with singularities. In this case, one can apply the space decomposition principle, but this is of little practical use for a general three-dimensional domain, because the spaces of singularities lacks so far a practical description, even though some interesting charcterisations of them are known.

On the other hand, when symmetry considerations allow to reduce the equations to two-dimensional problems, more precise results can be proven. They are also of the type C^\alpha(0,T;H^s(\Omega)), where the limiting exponents \alpha and s are controlled by the geometry of the singularities. These developments rely on known explicit expression for the singular fields, and on the singularity theory of the scalar wave equation by Grisvard.