The paper is a brief review on the existence and basic properties of static,
spherically symmetric regular black hole solutions of general relativity, where
the source of gravity is represented by nonlinear electromagnetic fields with
the Lagrangian function $L$ depending on the single invariant $f =
F_{\mu\nu}F^{\mu\nu}$ or on two variables: either $L(f, h)$, where $h =
{^*}F_{\mu\nu} F^{\mu\nu}$, where ${^*}F_{\mu\nu}$ is the Hodge dual of
$F_{\mu\nu}$, or $L(f, J)$, where $J = F_{\mu\nu}F^{\nu\rho} F_{\rho\sigma}
F^{\sigma\mu}$. A number of no-go theorems are discussed, revealing the
conditions under which the space-time cannot have a regular center, among which
the theorems concerning $L(f,J)$ theories are probably new. These results
concern both regular black holes and regular particlelike or starlike objects
(solitons) without horizons. Thus, a regular center in solutions with an
electric charge $q_e\ne 0$ is only possible with nonlinear electrodynamics
(NED) having no Maxwell weak field limit. Regular solutions with $L(f)$ and
$L(f, J)$ NED, possessing a correct (Maxwell) weak-field limit, are possible if
the system contains only a magnetic charge $q_m \ne 0$. It is shown, however,
that in such solutions the causality and unitarity as well as dynamic stability
conditions are inevitably violated in a neighborhood of the center. Some
particular examples are discussed.
