  The regularized Maxwell theory is a recently discovered theory of non-linear
electrodynamics that admits many important gravitating solutions within the
Einstein theory. Namely, it was originally derived as the unique non-linear
electrodynamics (that depends only on the field invariant
$F_{\mu\nu}F^{\mu\nu}$) whose radiative solutions can be found in the
Robinson--Trautman class. At the same time, it is the only electrodynamics of
this type (apart from Maxwell) whose slowly rotating solutions are fully
characterized by the electrostatic potential. In this paper, after discussing
the basic properties of the regularized Maxwell theory, we concentrate on its
spherical electric solutions. These not only provide `the simplest'
regularization of point electric field and its self-energy, but also feature
complex thermodynamic behavior (in both canonical and grandcanonical ensembles)
and admit an unprecedented phase diagram with multiple first-order,
second-order, and zeroth-order phase transitions. Among other notable
solutions, we construct a novel C-metric describing accelerated AdS black holes
in the regularized Maxwell theory. We also present a generalization of the
regularized Maxwell Lagrangian applicable to magnetic solutions, and find the
corresponding spherical, slowly rotating, and weakly NUT charged solutions.
