The "standard conjectures" in Kahler geometry state that the existence of a canonical metric in a given Hodge class is equivalent to the stability of the associated projective model(s). There are several competing definitions of stability ( mainly due to Tian and Donaldson ) and various partial results linking these definitions to the sought after metric. I will give a survey/progress report of my own work on this problem. The reference for the talk is: http://arxiv.org/pdf/0811.2548v3