Hodge structure
A fundamental aspect of complex geometry is that there are continuous families of non-isomorphic complex manifolds (which are all diffeomorphic as real manifolds). Phillip Griffiths's notion of a variation of Hodge structure describes how the Hodge structure of a smooth complex projective variety "X" varies when "X" varies. In geometric terms, this amounts to studying the period mapping associated to a family of varieties. Saito's theory of Hodge modules is a generalization. Roughly speaking, a mixed Hodge module on a variety "X" is a sheaf of mixed Hodge structures over "X", as would arise from a family of varieties which need not be smooth or compact.