  The endomorphism algebra of a K3 type Hodge structure is a totally real field
or a CM field. In this paper we give a low brow introduction to the case of a
totally real field. We give existence results for the Hodge structures, for
their polarizations and for certain K3 surfaces. We consider the Kuga Satake
variety of these Hodge structures and we discuss some examples. Finally we
indicate various open problems related to the Hodge conjecture.
