  We prove that the bounded derived category of coherent sheaves on a smooth
projective complex variety reconstructs the isomorphism classes of fibrations
onto smooth projective curves of genus $g\geq 2$. Moreover, in dimension at
most four, we prove that the same category reconstructs the isomorphism classes
of fibrations onto normal projective surfaces with positive holomorphic Euler
characteristic and admitting a finite morphism to an abelian variety. Finally,
we study the derived invariance of a class of fibrations with minimal
base-dimension under the condition that all the Hodge numbers of type
$h^{0,p}(X)$ are derived invariant.
