Finding geometrical correspondences between two images, called image registration, is one of the numerous challenging problems in image processing. Commonly, image registration is phrased as a variational problem that is known to be ill-posed. Thus, regularization is used to ensure existence of solutions, introduce prior knowledge about the expected solution, and/or increase the robustness against noise. This talk gives a comprehensive overview of theory, numerical methods, and applications of regularization energies based on hyperelasticity.