In its original form, the Shafarevich hyperbolicity theorem considers a smooth, projective family of algebraic curves over a smooth quasi-projective base curve Y. It asserts that if Y is of special type, then the family is necessarily isotrivial. The lectures discuss hyperbolicity properties of moduli stacks and generalisations of Shafarevich hyperbolicity to higher dimensions. They concentrates on methods and results that relate moduli theory with recent progress in higher dimensional birational geometry.