In the 1920s, building on Fermat's Last Theorem, Mordell conjectured that the set of rational points of any smooth projective curve of genus at least two, over any number field, is finite. In the 1960s, Shafarevich turned this into a purely algebro-geometric conjecture involving families of smooth projective curves. Parshin, Arakelov and Faltings settled this conjecture by showing that the base spaces of such families are in some sense hyperbolic, as long as there is some variation in the algebraic structure of the fibers. Inspired by recent advances in Birational Geometry, Kebekus and Kovács conjectured that these hyperbolicity-type properties should hold for a vast class of projective families, with fibers of arbitrary dimension. In this talk I will discuss this conjecture and my solution to it. I will also talk about more recent advances in this area, based on a joint work with Kovács (University of Washington).