  We study $D$-dimensional charged static spherically symmetric black hole
solutions in Gauss-Bonnet theory coupled to nonlinear electrodynamics defined
as arbitrary functions of the field invariant and constrained by several
physical conditions. These solutions are characterized in terms of the mass
parameter $m$, the electromagnetic energy $\varepsilon$ and the Gauss-Bonnet
parameter $l_{\alpha}^2$. We find that a general feature of these solutions is
that the metric behaves in a different way in $D=5$ and $D>5$ space-time
dimensions. Moreover, such solutions split into two classes, according to
whether they are defined everywhere or show branch singularities, depending on
($m, \varepsilon, l_{\alpha}^2$). We describe qualitatively the structures
comprised within this scenario, which largely extends the results obtained in
the literature for several particular families of nonlinear electrodynamics. An
explicit new example, illustrative of our results, is introduced. Finally we
allow non-vanishing values of the cosmological constant length $l_{\Lambda}^2$,
and study the existence of new structures, in both asymptotically Anti-de
Sitter and de Sitter spaces.
