The Calabi problem asks which manifolds can be endowed with a canonical metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation - a so-called Kähler-Einstein metric. The Yau Tian-Donaldson conjecture (now a theorem) states that a Fano manifold admits a Kähler-Einstein metric precisely when it satisfies a sophisticated algebraic condition called K-polystability.

Surprisingly, the notion of K-polystability also sheds some light on another poorly understood aspect of the geometry of Fano varietieshow they behave in families.

We have known for a long time that the set of all Fano varieties does not form a reasonable moduli space, but recent works have shown that the set of K-polystable Fano varieties does, and these are called K-moduli spaces.

Few examples of K-moduli spaces are known. In dimension 3, the classification of Fano manifolds into 105 deformation families is an invitation to investigate K-moduli spaces of those families with K- polystable members.

In this talk, I will discuss K-moduli spaces of Fano 3-folds, and describe explicitly some of their components of small dimension.