We consider the problem of constructing genus 3 curves defined over the complex field with the property that their Jacobians are simple and admit complex multiplication (CM). In genus 1 and 2, a natural answer comes from the theory of complex multiplication of Shimura and Taniyama, since all simple principally polarized abelian varieties (p.p.a.v.) of dimension 1 and 2 are isomorphic to Jacobians of hyperelliptic curves. In genus 3, the situation is more complicated and thus more interesting. Up to isomorphism, every p.p.a.v. of dimension 3 is the Jacobian of a complete smooth projective curve of genus 3. Moreover, this curve is isomorphic either to a hyperelliptic or a plane quartic curve. However, the moduli space of p.p.a.v. in dimension 3 has dimension 6 and the subspace of Jacobians of hyperelliptic curves has codimension 1. Therefore, given a random sextic CM field, we expect that the set of p.p.a.v. with CM by the maximal order of that field will not contain any hyperelliptic Jacobians. We will review progress made in recent years on the construction of Jacobians of genus 3 hyperelliptic curves with CM.