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Meshes generated by range scanners and other acquisition tools are often incomplete and typically contain multiple connected components with irregular boundaries and complex holes. This paper introduces a robust algorithm for completion of such meshes using a mapping between the incomplete mesh and a template model. The mapping is computed using a novel framework for bijective parameterization of meshes with gaps and holes. We employ this mapping to correctly glue together the components of the input mesh and to close the holes. The template is used to fill in the topological and geometric information missing in the input. The completed models are guaranteed to have the same topology as the template. Furthermore, if no appropriate template exists or if only topologically correct completion is required a standard canonical shape can be used as a template. As part of our completion method we propose a boundary-mapping technique useful for mesh editing operations such as merging, blending, and detail transfer. We demonstrate that by using this technique we can automatically perform complex editing operations that previously required a large amount of user interaction.