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This survey reviews the recent advances in linear variational mesh deformation techniques. These methods were developed for editing detailed high-resolution meshes like those produced by scanning real-world objects. The challenge of manipulating such complex surfaces is threefold: The deformation technique has to be sufficiently fast, robust, intuitive, and easy to control to be useful for interactive applications. An intuitive and, thus, predictable deformation tool should provide physically plausible and aesthetically pleasing surface deformations, which, in particular, requires its geometric details to be preserved. The methods that we survey generally formulate surface deformation as a global variational optimization problem that addresses the differential properties of the edited surface. Efficiency and robustness are achieved by linearizing the underlying objective functional such that the global optimization amounts to solving a sparse linear system of equations. We review the different deformation energies and detail preservation techniques that were proposed in recent years, together with the various techniques to rectify the linearization artifacts. Our goal is to provide the reader with a systematic classification and comparative description of the different techniques, revealing the strengths and weaknesses of each approach in common editing scenarios.