Abstract: Surface reconstruction algorithms build digital models of real world objects from data recorded by a scanning device. Since scanning devices are not perfect, they introduce noise and outliers into the recorded data. From such noisy data, an effective reconstruction algorithm must produce models that reflect the geometry and the topology of the sampled surface. In this thesis we analyze surface reconstruction algorithms and describe a software system that we have developed for building three-dimensional models of real world objects. Implicit methods for surface reconstruction are widely used in computer graphics as they are fast, easy to implement, and scale well to large point clouds. However, these implicit methods come with no provable guarantees on the reconstructed surface. We analyze an implicit surface reconstruction algorithm based on a data interpolation technique called moving least squares (MLS). We prove that under certain sampling conditions, the reconstructed surface is an accurate geometric and topological representation of the original surface. Our sampling requirements are adaptive and allow for noise in the input data set. Delaunay-based surface reconstruction algorithms build the reconstructed surface as a set of triangles from the Delaunay tetrahedralization of the sample points. Many Delaunay-based reconstruction algorithms have been proposed with guarantees on the reconstructed surface when the input point cloud satisfies certain sampling conditions. However, most point clouds obtained from scanning devices violate these sampling conditions. We present a Delaunay-based reconstruction algorithm called eigencrust that uses spectral partitioning to robustly deal with noise, outliers, and regions of undersampling in the input point cloud. We show empirical evidence that our implementation of eigencrust is substantially more robust than several closely related surface reconstruction programs. We describe a software system that we have developed for reconstructing three-dimensional models from data recorded using range scanners. Range images from a scanning device are automatically aligned using point signatures called harmonic shape contexts. An implementation of the MLS algorithm defines a smooth surface approximating the scanned surface. Finally, an eigencrust implementation meshes the surface defined by the MLS algorithm.