The moduli space $A_g$ of complex principally polarized Abelian varieties of dimension $g$ is an irreducible quasi-projective variety, which, for $g>1$, is neither affine nor projective. One can measure how far $A_g$ is from being affine/projective through looking at the largest dimension of a compact complex subvariety, or rather at the largest dimension of a compact complex subvariety passing through a very general point of $A_g$. In a joint paper with S. Grushevsky, R. Salvati Manni and J. Tsimerman, we determine both these values and in fact we characterize compact complex subvarieties of $A_g$ of largest dimension. In this talk I will describe the main ideas and techniques used in the above calculation.