Much effort is invested in generating natural deformations of three-dimensional shapes. Deformation transfer simplifies this process by allowing to infer deformations of a new shape from existing deformations of a similar shape. Current deformation transfer methods can be applied only to shapes which are represented as a single component manifold mesh, hence their applicability to real-life 3D models is somewhat limited. We propose a novel deformation transfer method, which can be applied to a variety of shape representations - tet-meshes, polygon soups and multiple-component meshes. Our key technique is deformation of the space in which the shape is embedded. We approximate the given source deformation by a harmonic map using a set of harmonic basis functions. Then, given a sparse set of user-selected correspondence points between the source and target shapes, we generate a deformation of the target shape which has differential properties similar to those of the source deformation. Our method requires only the solution of linear systems of equations, and hence is very robust and efficient. We demonstrate its applicability on a wide range of deformations, for different shape representations.