Given a family of projective manifolds, Hodge theory provides a way to study the variation of complex structures of the family of manifolds. Using the flat structure associated to their homology groups, one can construct certain locally defined period integrals, as a means to parametrize the complex structures. These integrals, however, depend on a number of choices in general which render them difficult to compute.

Thanks to Mirror Symmetry, a large class of distinguished families of projective manifolds -- known as Calabi-Yau manifolds -- have been found in recent years, where the period integrals can be constructed in a canonical fashion. In this topic course, this construction will be explained and given a new interpretation, which leads us to a vast generalization. In examples, we will give an explicit construction for a new class of partial differential equation systems that govern those period integrals.