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We study K3 surfaces with complex multiplication following the classical workof Shimura on CM abelian varieties. After we translate the problem in terms ofthe arithmetic of the CM field and its id\`{e}les, we proceed to study someabelian extensions that arise naturally in this context. We then make use ofour computations to determine the fields of moduli of K3 surfaces with CM andto classify their Brauer groups. More specifically, we provide an algorithmthat given a number field $K$ and a CM number field $E$, returns a finite listsof groups which contains $\mathrm{Br}(\overline{X})^{G_K}$ for any K3 surface$X/K$ that has CM by the ring of integers of $E$. We run our algorithm when $E$is a quadratic imaginary field (a condition that translates into $X$ havingmaximal Picard rank) generalizing similar computations already appearing in theliterature.

Complex multiplication and Brauer groups of K3 surfaces