This paper is a survey of some recent progress on the study of Calabi's
extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture
relating the existence of extremal metrics to an algebro-geometric stability
notion and we give some example settings where this conjecture has been
established. We then turn to the question of what one expects when no extremal
metric exists.Comment: 17 pages, 4 figures. Contribution to the proceedings of the 2014 IC