Projective hyper-Kähler (HK) manifolds are among the building blocks of projective manifolds with trivial first Chern class. Fano manifolds are projective manifolds with positive first Chern class.

Despite the fact that these two classes of algebraic varieties are very different (HK manifolds have a holomorphic symplectic form which governs all of its geometry, Fano manifolds have no holomorphic forms) their geometries have some strong ties. For example, starting from some special Fano manifolds one can sometimes construct HK manifolds as parameter spaces of objects on the Fano. In this talk I will explain this circle of ideas and focus on some recent work exploring the converse: given a projective HK manifold, how to recover a Fano manifold from it?