  General Relativity has shown an outstanding observational success in the
scales where it has been directly tested. However, modifications have been
intensively explored in the regimes where it seems either incomplete or signals
its own limit of validity. In particular, the breakdown of unitarity near the
Planck scale strongly suggests that General Relativity needs to be modified at
high energies and quantum gravity effects are expected to be important. This is
related to the existence of spacetime singularities when the solutions of
General Relativity are extrapolated to regimes where curvatures are large. In
this sense, Born-Infeld inspired modifications of gravity have shown an
extraordinary ability to regularise the gravitational dynamics, leading to
non-singular cosmologies and regular black hole spacetimes in a very robust
manner and without resorting to quantum gravity effects. This has boosted the
interest in these theories in applications to stellar structure, compact
objects, inflationary scenarios, cosmological singularities, and black hole and
wormhole physics, among others. We review the motivations, various
formulations, and main results achieved within these theories, including their
observational viability, and provide an overview of current open problems and
future research opportunities.
