In '57 Calabi posed a conjecture about the existence of canonical metrics on Kähler manifolds that became a central problem on Complex Geometry. This conjecture was solved by Yau in 78. Not long after many attempts to generalize it to the Hermitian setting began but only in 2010 Tossati-Weinkove were able to completely solve it for Hermitian manifolds.
In this talk we will present an overview of the long proof of the Hermitian version of the Calabi conjecture, its connection to the Complex Monge-Ampère equation and highlight differences from the Kähler case. The techniques presented on the talk will differ from those in the original proof, more recently developed techniques of Pluripotential Theory will be used to give a few details of the proof as time allows.