  We investigate versions of the Shafarevich conjecture, as proved for curves
and abelian varieties by Faltings, for other classes of varieties. We first
obtain analogues for certain Fano threefolds. We use these results to prove the
Shafarevich conjecture for smooth sextic surfaces, which appears to be the
first non-trivial result in the literature on the arithmetic of such surfaces.
Moreover, we exhibit certain moduli stacks of Fano varieties which are not
hyperbolic, which allows us to show that the analogue of the Shafarevich
conjecture does not always hold for Fano varieties. Our results also provide
new examples for which the conjectures of Campana and Lang-Vojta hold.
