Compact Kähler manifolds are transcendental generalizations of complex projective manifolds, bearing a special metric structure. Inside the large class of compact Kähler manifolds with non-positive curvature, many aspects of such manifolds (topological, metric, geometric) are now reasonably well understood. However, the study of their moduli spaces or the search for minimal models in birational geometry have made it imperative to extend our understanding to non-compact or singular varieties. In that direction, we propose to investigate a wide array of questions using both analytic and algebraic methods, among which: - The uniformization problem - Analyze the behavior of canonical metrics at the boundary and construct explicit examples - Study the geometry of special subvarieties of moduli spaces or quotients of bounded symmetric domains - The Iitaka conjecture in the non-positive curvature case - Study hyperbolicity of base spaces of singular Calabi-Yau fibrations