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We present a GPU-based framework for extracting surface and curve skeletons of 3D shapes represented as large polygonal meshes. We use an efficient parallel search strategy to compute point-cloud skeletons and their distance and feature transforms (FTs) with user-defined precision. We regularize skeletons by a new GPU-based geodesic tracing technique which is orders of magnitude faster and more accurate than comparable techniques. We reconstruct the input surface from skeleton clouds using a fast and accurate image-based method. We also show how to reconstruct the skeletal manifold structure as a polygon mesh and the curve skeleton as a polyline. Compared to recent skeletonization methods, our approach offers two orders of magnitude speed-up, high-precision, and low-memory footprints. We demonstrate our framework on several complex 3D models.