Subvarieties of Grassmannians (and especially Fano varieties) obtained
from section of homogeneous vector bundles are far from being
classified. A case of particular interest is given by the Fano varieties
of K3 type (FK3), for their deep links with hyperkähler geometry. In
this talk we will present some examples of recently discovered FK3
varieties, and a general procedure that allows us to spread a (Hodge) K3
structure as a component of the Hodge structure of different varieties.
This is in collaboration with Giovanni Mongardi and Marcello