Explore complex algebraic geometry in this 1-hour lecture focusing on realizing moduli spaces as locally symmetric varieties through Hodge theory. Survey existing results and ideas, including examples like moduli of polarized abelian varieties, Deligne-Mostow theory, and moduli of cubic hypersurfaces. Delve into work with Zhiwei Zheng on families of Calabi-Yau manifolds with periods in complex hyperbolic balls. Gain insights into this important topic and its applications in understanding special geometric objects.