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The study of the Kodaira dimension of moduli spaces is a classical topic in geometry, which is related in interesting ways with the geometry of the objects which are parametrized by the moduli space. In a joint work with I. Barros, E. Brakkee and L. Flapan, we use techniques of Gritsenko-Hulek-Sankaran involving the Borcherds modular form to determine a bound on the degree of the polarization beyond which the moduli spaces of some polarized hyperkähler manifolds are all of general type. In this talk we explain the strategy we followed and we determine explicitly our bound in some cases. We also present new examples of unirational moduli spaces of polarized hyperkähler manifolds.