This paper presents a new algorithm for surface reconstruction from unorganized sample points in 3D space. The algorithm is the first to provide provable guarantees for surface reconstruction. It is based on the three-dimensional Voronoi diagram and Delaunay triangulation. The algorithm produces a set of triangles called the "crust" that interpolates the input points rather than approximating them. The output mesh is guaranteed to be topologically correct and converges to the original surface as the sampling density increases. The algorithm defines a "good sample" as one where the sampling density is inversely proportional to the distance to the medial axis of the surface. This allows for locally varying sampling densities, with denser sampling near features and sparser sampling in featureless areas. The algorithm requires no parameters, as it automatically computes the necessary sampling density locally. The algorithm works by computing the Voronoi diagram of the sample points and using the Delaunay triangulation of the sample points and Voronoi vertices to construct the crust. The crust consists of triangles that are guaranteed to be close to the original surface. The algorithm also includes a normal filtering step to ensure that the output mesh converges in surface normal as the sampling density increases. The algorithm is implemented using a freely available exact-arithmetic Voronoi diagram code, and is efficient enough to handle 10,000 points in a matter of minutes. The main difficulty in practice is the reconstruction of sharp edges, which requires additional heuristics to handle. The algorithm is compared to other surface reconstruction methods, including the α-shape and zero-set algorithms. It differs from these methods in that it does not require a parameter, and it is more suitable for surface reconstruction than for determining the natural dimensionality of a point set. The algorithm has been tested on a variety of data sets, including medical imagery, laser range scanners, and mathematical models. It has been shown to produce visually acceptable results, and to be robust to noise and undersampling. The algorithm is also applicable to the reconstruction of surfaces with sharp edges and boundaries, and can be used for lossless mesh compression by representing the surface entirely by its vertices.This paper presents a new algorithm for surface reconstruction from unorganized sample points in 3D space. The algorithm is the first to provide provable guarantees for surface reconstruction. It is based on the three-dimensional Voronoi diagram and Delaunay triangulation. The algorithm produces a set of triangles called the "crust" that interpolates the input points rather than approximating them. The output mesh is guaranteed to be topologically correct and converges to the original surface as the sampling density increases. The algorithm defines a "good sample" as one where the sampling density is inversely proportional to the distance to the medial axis of the surface. This allows for locally varying sampling densities, with denser sampling near features and sparser sampling in featureless areas. The algorithm requires no parameters, as it automatically computes the necessary sampling density locally. The algorithm works by computing the Voronoi diagram of the sample points and using the Delaunay triangulation of the sample points and Voronoi vertices to construct the crust. The crust consists of triangles that are guaranteed to be close to the original surface. The algorithm also includes a normal filtering step to ensure that the output mesh converges in surface normal as the sampling density increases. The algorithm is implemented using a freely available exact-arithmetic Voronoi diagram code, and is efficient enough to handle 10,000 points in a matter of minutes. The main difficulty in practice is the reconstruction of sharp edges, which requires additional heuristics to handle. The algorithm is compared to other surface reconstruction methods, including the α-shape and zero-set algorithms. It differs from these methods in that it does not require a parameter, and it is more suitable for surface reconstruction than for determining the natural dimensionality of a point set. The algorithm has been tested on a variety of data sets, including medical imagery, laser range scanners, and mathematical models. It has been shown to produce visually acceptable results, and to be robust to noise and undersampling. The algorithm is also applicable to the reconstruction of surfaces with sharp edges and boundaries, and can be used for lossless mesh compression by representing the surface entirely by its vertices.