  Seminar held at JINR, Dubna, May 15, 2012.
  In General Relativity, spacetime singularities raise a number of problems,
both mathematical and physical. One can identify a class of singularities -
with smooth but degenerate metric - which, under a set of conditions, allow us
to define proper geometric invariants, and to write field equations, including
equations which are equivalent to Einstein's at non-singular points, but remain
well-defined and smooth at singularities. This class of singularities is large
enough to contain isotropic singularities, warped-product singularities,
including the Friedmann-Lemaitre-Robertson-Walker singularities, etc. Also a
Big-Bang singularity of this type automatically satisfies Penrose's Weyl
curvature hypothesis.
  The Schwarzschild, Reissner-Nordstrom, and Kerr-Newman singularities
apparently are not of this benign type, but we can pass to coordinates in which
they become benign. The charged black hole solutions Reissner-Nordstrom and
Kerr-Newman can be used to model classical charged particles in General
Relativity. Their electromagnetic potential and electromagnetic field are
analytic in the new coordinates - they have finite values at r=0. There are
hints from Quantum Field Theory and Quantum Gravity that a dimensional
reduction is required at small scale. A possible explanation is provided by
benign singularities, because some of their properties correspond to a
reduction of dimensionality.
