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The survivable network design problem is a classical problem in combinatorial optimization of constructing a minimum-cost subgraph satisfying predetermined connectivity requirements. In this paper we consider its geometric version in which the input is a complete Euclidean graph. We assume that each vertex v has been assigned a connectivity requirement rv. The output subgraph is supposed to have the vertex- (or edge-, respectively) connectivity of at least min{rv, ru} for any pair of vertices v, u.We present the first polynomial-time approximation schemes (PTAS) for basic variants of the survivable network design problem in Euclidean graphs. We first show a PTAS for the Steiner tree problem, which is the survivable network design problem with rv 驴 {0, 1} for any vertex v. Then, we extend it to include the most widely applied case where rv 驴 {0, 1, 2} for any vertex v. Our polynomial-time approximation schemeswork for both vertex- and edge-connectivity requirements in time O(n log n), where the constants depend on the dimension and the accuracy of approximation. Finally, we observe that our techniques yield also a PTAS for the multigraph variant of the problem where the edge-connectivity requirements satisfy rv 驴 {0, 1, . . . , k} and k = O(1).