This is a report on joint work with Behrouz Taji. Given a flat projective morphism f from X to B of complex varieties, assuming that B is smooth, we construct a system of reflexive Hodge sheaves on B. If in addition X is also smooth then this system gives an extension of the Hodge bundle underlying the VHS of the smooth locus of f. This in turn provides a criterion that all VHSs of geometric origin must satisfy. As an independent application we prove a singular version of Viehweg's conjecture about base spaces of families of maximal variation. Speaker(s): Sándor Kovács (University of Washington)