  We study abelian varieties and K3 surfaces with complex multiplication
defined over number fields of fixed degree. We show that these varieties fall
into finitely many isomorphism classes over an algebraic closure of the field
of rational numbers. As an application we confirm finiteness conjectures of
Shafarevich and Coleman in the CM case. In addition we prove the uniform
boundedness of the Galois invariant subgroup of the geometric Brauer group for
forms of a smooth projective variety satisfying the integral Mumford--Tate
conjecture. When applied to K3 surfaces, this affirms a conjecture of
V\'arilly-Alvarado in the CM case.
