  Around 1970 Griffiths introduced the moduli of polarized Hodge structures/the
period domain $D$ and described a dream to enlarge $D$ to a moduli space of
degenerating polarized Hodge structures. Since in general $D$ is not a
Hermitian symmetric domain, he asked for the existence of a certain automorphic
cohomology theory for $D$, generalizing the usual notion of automorphic forms
on symmetric Hermitian domains. Since then there have been many efforts in the
first part of Griffith's dream but the second part still lives in darkness. The
objective of the present text is two-folded. First, we give an exposition of
the subject. Second, we give another formulation of the Griffiths problem,
based on the classical Weierstrass uniformization theorem.
