  We consider spherically symmetric configurations in general relativity,
supported by nonlinear electromagnetic fields with gauge-invariant Lagrangians
depending on the single invariant $f = F_{\mu\nu} F^{\mu\nu}$. Static black
hole and solitonic solutions are briefly described, both with only an electric
or magnetic charge and with both nonzero charges (the dyonic ones). It is
stressed that only pure magnetic solutions can be completely nonsingular. For
dyonic systems, apart from a general scheme of obtaining solutions in
quadratures for an arbitrary Lagrangian function $L(f)$, an analytic solution
is found for the truncated Born-Infeld theory (depending on the invariant $f$
only). Furthermore, considering spherically symmetric metrics with two
independent functions of time, we find a natural generalization of the class of
wormholes found previously by Arellano and Lobo with a time-dependent conformal
factor. Such wormholes are shown to be only possible for some particular
choices of the function $L(f)$, having no Maxwell weak-field limit.
