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The aim of this paper is to estimate the irrationality of moduli spaces ofhyperk\"ahler manifolds of types K3$^{[n]}$, Kum$_{n}$, OG6, and OG10. We provethat the degrees of irrationality of these moduli spaces are bounded from aboveby a universal polynomial in the dimension and degree of the manifolds theyparametrize. We also give a polynomial bound for the degrees of irrationalityof moduli spaces of $(1,d)$-polarized abelian surfaces.

On the irrationality of moduli spaces of projective hyperk\"ahler manifolds