This survey is devoted to discussion of problems of the unique determination of conformal type for domains in Euclidean spaces. It consists of three parts. In the first of them, we consider results on the problem of the unique determination for (generally speaking) nonconvex domains in Rn, where n ≥ 4. The second part is devoted to study of problems on unique determination of convex polyhedral domains in three-dimensional Euclidean space by relative conformal moduli of boundary condensers. Finally, in the third part, we study problems of unique determination of 3-connected plane domains by relative conformal moduli of pairs of boundary components.