Let G be a complex reductive group and C be a complex smooth projective curve of genus at least two. The moduli space of G-Higgs bundles over C has a natural structure of holomorphic symplectic quasi-projective variety. Furthermore, it admits a proper morphism (the Hitchin morphism) onto an affine space (the Hitchin basis). It turns out that the Hitchin morphism is a Lagrangian fibration whose general fiber is an abelian variety. In this talk, we provide a description of the locus of the singular fibers of the Hitchin morphism. As an application, we present a proof of a Torelli-type theorem for the moduli space of G-bundles over C.