This survey is devoted to discussing the problems of the unique determination of surfaces that are the boundaries of (generally speaking) nonconvex domains. First (in Sec. 2) we examine some results on the problem of the unique determination of domains by the relative metrics of the boundaries. Then, in Sec. 3, we study rigidity conditions for the boundaries of submanifolds in a Riemannian manifold. The final part (Sec. 4) is concernedwith the unique determination of domains by the condition of the local isometry of boundaries in the relative metrics.