  This is a review article discussing the de Rham cohomology of period domains
of Hodge structures. We explain it as the de Rham cohomology of differentiable
stacks as of a moduli space. We also discuss the cohomology of the partial
toroidal compactification of these domains using known formulas on cohomology
or Chow rings of toric structures. The text is expository and we have tried to
connect some existing ideas that probably their relations not processed in the
literature of Hodge theory. We state the significance of ideas as they
naturally could be related, probably with not serious mathematical proof. The
proofs stated in the text maybe expressed in a more serious context.
