  We study in this article three birational invariants of projective
hyper-K\"ahler manifolds: the degree of irrationality, the fibering gonality
and the fibering genus. We first improve the lower bound in a recent result of
Voisin saying that the fibering genus of a Mumford--Tate very general
projective hyper-K\"ahler manifold is bounded from below by a constant
depending on its dimension and the second Betti number. We also study the
relations between these birational invariants for projective K3 surfaces of
Picard number 1 and study the asymptotic behaviors of their degree of
irrationality and fibering genus.
