  This work presents the foundations of Singular Semi-Riemannian Geometry and
Singular General Relativity, based on the author's research. An extension of
differential geometry and of Einstein's equation to singularities is reported.
Singularities of the form studied here allow a smooth extension of the Einstein
field equations, including matter. This applies to the Big-Bang singularity of
the FLRW solution. It applies to stationary black holes, in appropriate
coordinates (since the standard coordinates are singular at singularity, hiding
the smoothness of the metric). In these coordinates, charged black holes have
the electromagnetic potential regular everywhere. Implications on Penrose's
Weyl curvature hypothesis are presented. In addition, these singularities
exhibit a (geo)metric dimensional reduction, which might act as a regulator for
the quantum fields, including for quantum gravity, in the UV regime. This opens
the perspective of perturbative renormalizability of quantum gravity without
modifying General Relativity.
