  We use the theory of Hodge modules to construct Viehweg-Zuo sheaves on the
base spaces of families with maximal variation and fibers of general type, or
more generally whose geometric generic fiber has a good minimal model. We
deduce Viehweg's hyperbolicity conjecture in this context, namely the fact that
the base spaces of such families are of log general type. This is approached as
part of a general problem of identifying what spaces can support Hodge
theoretic objects with certain positivity properties.
