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Hodge Theory provides a general way of understanding moduli spaces of algebraic varieties: Given a family of algebraic varieties, one obtains a `period map' by considering the hodge structure on the cohomology. However, these period maps are built from period integrals, and are highly transcendental, which yields challenges when one wants to recover an algebraic structure. 

Famously, this story works really nicely for the moduli space of (principally polarized, g-dimensional) Abelian varieties A_g, where the hodge theory gives an exact moduli space, and the work of Baily-Borel provides a beautiful compactification of this space which can also be understood using hodge theory. 

We explain how this picture generalizes to arbitrary period maps. This has especially nice applications to moduli spaces of Calabi-Yaus, which has proven less accessible to other techniques. Moreover, the same tools yield a resolution of the  b-semiampleness conjecture of Prokhorov and Shokurov. This is joint work with Bakker, Filipazzi, and Mauri.