The Calabi problem asks which compact complex manifolds are Kähler-Einstein (KE)- i.e. can be endowed with a canonical metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation. Such manifolds always have a canonical class of definite sign. The existence of such a metric on manifolds with positive and trivial canonical class (general type and Calabi Yau) was proved in the 70s. In the case of Fano manifolds, the situation is more subtle - Fano manifolds may or may not have a KE metric. 

We now know that a Fano manifold admits a KE metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, K-polystability also sheds some light on another important problem: while Fano varieties do not behave well in families, K-polystable Fano varieties do and form K-moduli spaces. 

Our explicit understanding of K-polystability is still partial, and few examples of K-moduli spaces are known. In dimension 3, Fano manifolds were classified into 105 deformation families by Mori-Mukai and Iskovskikh. 

 In this talk, I will present an overview of the Calabi problem and discuss its solution in dimension 3. Knowing - as we do - which families in the classification of Fano 3-folds have K-polystable members is a starting point to investigate the corresponding K-moduli spaces. I will describe explicitly some K-moduli spaces of Fano 3-folds of small dimension.