Let X be a complex projective K3 surface having complex multiplication by a CM field E , and let T_{X} be its transcendental lattice. We say that X has maximal complex multiplication if \mathrm{End}{\mathrm{Hdg}}(T{X}) is the ring of integers of E .For which CM fields E does such a K3 surface exist? What are the possibilities for the transcendental lattices, Picard lattices of these surfaces? The aim of this paper is to study these questions and give some examples.