  We introduce the notion of a variety (or more generally a motive) of CM-type
which generalises the well known notion of abelian variety of CM-type. Just as
in that particular case it will turn out that the cohomology of the variety is
determined by purely combinatorial data; the type of the variety. As
applications we will show that the \l-adic representations are given by
algebraic Hecke characters whose algebraic parts are determined by the type and
give a method for computing the discriminant of the N\'eron-Severi group of
super-singular Fermat surfaces.
