  We show the existence of metrically dense entire curves in rationally
connected complex projective manifolds confirming for this case a conjecture
according to which such entire curves on projective manifolds exist if and only
if these are "special". We also show that such a dense entire curve may be
chosen in such a way that it does not lift to any of its ramified covers,
answering in this case a question of Corvaja and Zannier about the Nevanlinna
analog of the `weak Hilbert property' of arithmetic geometry. We consider
briefly the other test case of the conjecture, namely manifolds with $c_1=0$.
Furthermore we discuss entire curves in normal rational surfaces avoiding the
singular locus.
