  We develop the theory of Griffiths period map, which relates the
classification of smooth projective varieties to the associated Hodge
structures, in the framework of Derived Algebraic Geometry. We complete the
description of the local period map as a morphism of derived deformation
functors, following the path marked by Fiorenza, Manetti and Martinengo. In the
end we show how to lift the local period map to a (non-geometric) morphism of
derived stacks, in order to construct a global version of that.
