Allowing the isomorphism class of algebraic varieties to vary in a family usually imposes strong conditions on the space parametrizing that family. This has been thoroughly studied for families of varieties whose canonical bundle is positive, leading to what is called the hyperbolicity of the moduli stack of such varieties. I will explain how recent advances in Hodge theory and in the study of holomorphic forms, facilitated especially by M. Saito's theory of Hodge modules, allow us to answer questions regarding the base spaces of more general (and conjecturally arbitrary) families of varieties. Speaker(s): Mihnea Popa (Northwestern University)