The moduli spaces of smooth curves of genus at least two and its compactification, the space of stable curves, are one of the most investigated objects of algebraic geometry. In the past two decades, natural higher dimensional generalizations, the moduli space of canonically polarized manifolds and of stable schemes have been constructed. After giving a short introduction to the above spaces, I will talk about their global geometry, in particular about the hyperbolicity properties of the moduli space of canonically polarized manifolds.