  This thesis deals with the problem of singularities in a family of extensions
of General Relativity in the Metric-Affine formalism. I introduce the
Metric-Affine formalism as a framework in which study extensions of GR. I
review its features and motivate it through its application in Bravais
crystals, where ideal crystals can be described through Riemannian formalism,
but a crystal with defects have to be described with in terms of an independent
connection. The simplest way to construct solutions different from GR in this
formalism is to take a quadratic gravity lagrangian with an electrovacuum
stress-energy tensor. This way, charged black hole solutions are obtained.
  The geometry of these new charged black hole solutions is analysed. Far from
the sources, the geometry of these new solutions is equal to the GR one up to
$1/r^4$ corrections. However, instead of a central singularity, they have a
wormhole structure, which can be completely regular or where the curvature
scalars diverge, depending on the mass and charge of the black hole. I study
the geodesics around the wormhole throat and conclude that the geodesics are
always complete even in the case of a curvature divergence. To analyse in more
detail the geometry, I also study the evolution of a scalar wave sent towards
the wormhole. The evolution is shown to be regular and I calculate the
transmission coefficients and transmission cross section for a naked wormhole.
  Finally, I study the same case for a Born-Infeld lagrangian in an arbitrary
number of dimension ($d>4$). In this setting, the wormhole structure is also
present and the geodesics are still complete, therefore showing that the
metric-affine formalism can resolve singularities for a variety of models, and
is not a particular feature of the quadratic Lagrangian for $d=4$.
