Author's introduction: Let \(H_g\) be the Siegel upper half space of genus \(g\), \(A_g=H_g/\mathrm{Sp}(2g, \mathbb Z)\). The purpose of this paper is to prove that for \(g\ge 9\), the moduli space \(A_g\) of principally polarized abelian varieties of genus \(g\) over \(\mathbb C\) is of general type. By the method of the toroidal compactifications, we can construct a projective variety \(\bar A_g\) such that \(\bar A_g - A_g\) has normal crossing and \(\bar A_g\) has only finite quotient singularities. Resolving these singularities, we get a projective non-singular model \(\tilde{A_g}\). We shall study the extension problems of pluri-canonical differential forms to \(\bar A_g\) and \(\tilde{A_g}\) and prove that there are sufficiently many pluri-canonical forms which extend to \(\tilde{A_g}\) such that the plurigenera \(P_k\) of \(\tilde{A_g}\) grows with the same order as \(k^{g(g+1)/2}\) for \(g\ge 9\), therefore \(A_g\) is of general type.