A variation of Hodge structure may be defined as an integral manifold of a
particular exterior differential system (EDS). Except in the classical case
when the period domain is Hermitian symmetric, the EDS and its integrability
conditions are non-trivial. The latter turns out to have non-classical
algebro-geometric consequences; one of these involves the Chern classes of
the Hodge bundles and another involves the co-dimension of Noether-Lefschetz
loci. These will be discussed in the talk. Slides for the talk are available at http://slides.msri2009.org/