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Abstract: In the 90's, Tian conjectured an analytic characterization of Kahler-Einstein metrics. This characterization can be viewed as a direct analogue of the celebrated Yamabe problem (for constant scalar curvature metrics in a conformal class) solved by combined work of Aubin, Trudinger and Schoen in the 80's. Tian also conjectured an analogue of the well-known Aubin-Moser-Trudinger inequality that is a cornerstone result in conformal geometry. In joint work with T. Darvas we disprove one of these conjectures, and prove the remaining ones. Our results extend to many other types of canonical metrics in Kahler geometry aside from Kahler-Einstein metrics. Somewhat surprisingly, the proof uses in an essential way techniques of metric space geometry applied to appropriate infinite-dimensional spaces.