This thesis deals with the study of the cohomology and the Kodaira dimension of some moduli spaces. In the first part we compute the intersection Betti numbers of the GIT models of two moduli spaces. They parametrize non-hyperelliptic Petri-general curves of genus four and numerically polarized Enriques surfaces of degree two respectively. In both cases, the strategy of the cohomological calculation relies on a general method developed by Kirwan to compute the cohomology of GIT quotients of projective varieties. This procedure is based on the equivariantly perfect stratification of the unstable points studied by Hesselink, Kempf, Kirwan and Ness, and a partial resolution of singularities, called the Kirwan blow-up. In the second part of the thesis, we study the moduli spaces of elliptic K3 surfaces of Picard number at least three, i.e.\ $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\geq 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran.