In the 90's, Tian introduced a notion of properness in the space of Kahler metrics in terms of Aubin's nonlinear Dirichlet energy and Mabuchi's K-energy and put forward several conjectures on the relation between properness and existence of Kähler-Einstein metrics. These can be viewed as the Kahler analog of the classical Moser-Trudinger inequality from conformal geometry. In joint work with Y. Rubinstein we disprove one of these conjectures, and prove the remaining ones. Our techniques are flexible and extend to many different situations, including Kahler-Einstein edge metrics and Kahler-Ricci solitons. Moreover, we formulate a corresponding conjecture for constant scalar curvature metrics and reduce it to a PDE regularity problem of certain weak minimizers of the K-energy.