The Kodaira dimension of the moduli of K3 surfaces

The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46,50,54,58,60.