This book is an exposition of "Singular Semi-Riemannian Geometry"- the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. This book also dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry.The regularizations of singularities present in some Colombeau solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions and not classically.In this paper essentially new class Colombeau solutions to Einstein fild equations is obtained. We leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon.Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.However the Tolman formula for the total energy ET of a static and asymptotically flat spacetime,gives ET  m, as it should be. The vacuum energy density of free scalar quantum field  with a distributional background spacetime also is considered.It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such distributional BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes. In particular we obtain that the vacuum fluctuations 2  has a singular behavior on BHs horizon r: 2r~|r  r |2. A CHALLENGE TO THE BRIGHTNESS TEMPERATURE LIMIT OF THE QUASAR 3C273 explained successfully. Keywords: Colombeau nonlinear generalized funct