The Hilbert curve of a complex polarized manifold [Formula: see text] is the complex affine plane curve of degree [Formula: see text] defined by the Hilbert-like polynomial [Formula: see text], where [Formula: see text] is the canonical bundle of [Formula: see text] and [Formula: see text] and [Formula: see text] are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair [Formula: see text]. In particular, the existence of a fibration of [Formula: see text] over a variety of smaller dimension induced by a suitable adjoint bundle to [Formula: see text] translates into the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves of special varieties like Fano manifolds with low coindex, as well as fibrations over low-dimensional varieties having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, several polarized manifolds relevant for adjunction theory are completely characterized in terms of their Hilbert curves.