  This thesis covers several developments performed in metric-affine gravity.
This alternative framework extends General Relativity by considering a more
general connection than the one induced by the metric (i.e., arbitrary torsion
and nonmetricity participate in the dynamics). We start by revising some
mathematical aspects of the metric-affine framework, the way Lovelock and other
invariants are extended to it and the construction of a gauge formalism. Then
we focus on metric-affine pp-wave geometries and explore solutions of this type
for the quadratic metric-affine Lagrangian (with even-parity invariants).
Finally, in the last part of the manuscript, we analyze from a more
field-theoretical point of view the viability of different extensions of
General Relativity by guaranteeing the stability of their degrees of freedom.
