The aim of this minicourse is to explain the relation between polarized Hyperkähler manifolds, non-commutative K3 surfaces, and certain Fano manifolds.

In the first talk, I will introduce non-commutative K3 surfaces and moduli spaces of objects in them. The second talk will be centered on examples, including non-commutative K3 surfaces associated to cubic fourfolds and Gushel-Mukai manifolds; finally, we will discuss possible ways to associate a non-commutative K3 surface to a polarized Hyperkähler fourfold, and possible applications to Chow groups.