In  957 Eugenio Calabi   conjectured, that  a compact Kähler manifold  $M$ has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class of $M$.
The conjecture was solved  two decades later   by Shing-Tung Yau. by constructing a solution of the associated complex Monge-Ampère  equation using the continuity method.
In the seminar  I will give an overview  of the  Calabi-Yau Theorem and its generalizations in symplectic and Hermitian geometry.