  In the first part we survey some of the known results and conjectures on
compact Hyperkaehler (HK) manifolds. In the second part we presents a program
which aims to show that HK four-folds whose second cohomology (with 4-tuple
cup-product) is isomorphic to that of the Hilbert square of a K3 enjoy many of
the beautiful properties of K3 surfaces, in particular they should form a
single deformations class.
