On coupled Khler-Einstein metrics and weighted solitons on Fano manifolds

We consider coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds. These are natural generalizations of Kähler-Einstein metrics. As in the case of Kähler-Einstein metrics, the existence is known to be equivalent to algebraic conditions which generalize the K-polystability. In this survey, we outline recent developments for these two cases.