In this chapter we explain the basic construction methods of Calabi-yau manifolds with complex multiplication and give a first new example. We call a family of Calabi-Yau n-manifolds, which contains a dense set of fibers X such that the Hodge group of the Hodge structure on Hk(X,Q) is a torus for all k, a CMCY family of n-manifolds.
In Section 7.1 we explain the technical facts, which we will need for the construction of CMCY
families. By using the mirror construction of C. Borcea  and C. Voisin , we give a method to construct an infinite tower of CMCY
families in Section 7.2. In Section 7.3 we discuss the construction method of E. Viehweg and K. Zuo . By using this method given by a tower of cyclic covers, E. Viehweg and K. Zuo  have constructed an example of a CMCY
family of 3-manifolds. We finish this chapter with the example
$$P^3 \supset V(y_2^4 + y_1^4 + x_1 (x_1 - x_0 )(x_1 - \lambda x_0 )x_0 ) \to \lambda \in M_1$$
of a family of K
3 surfaces with a dense set of CM
fibers. This example is obtained from the Viehweg-Zuo tower, which starts with the family
$$P^2 \supset V(y_1^4 + x_1 (x_1 - x_0 )(x_1 - \lambda x_0 )x_0 ) \to \lambda \in M_1$$
of curves. This family has a dense set of CM
fibers by the previous chapter. By some of its involutions the family of K
3 surfaces above is suitable for the construction of a Borcea-Voisin tower.