One of the most ambitious goals of algebraic geometry is the classification of algebraic varieties. The first step in this ambitious plan is to classify varieties up to birational equivalence. The result of this first classification is that there are three basic classes of algebraic varieties starting from which all the other varieties can be constructed up to birational equivalence: canonical varieties, Calabi-Yau varieties and Fano varieties. They correspond to the case when the canonical line bundle is positive, null or negative.
The second step of this program is to construct moduli spaces of each of the above three classes of varieties, i.e. spaces that parametrize in a natural way all the varieties belonging to a fixed class.
The third and last step of the program is to compactify the resulting moduli spaces in a modular way, i.e. adding singular varieties in the boundary.
The aim of the seminar is to introduce this ambitious classification program and to present the state of the art, pointing out the main recent developments and the
many open problems.