Image registration is to automatically establish geometrical correspondences between two images. It is an essential task in almost all areas involving imaging. This chapter reviews mathematical techniques for nonlinear image registration and presents a general, unified, and flexible approach. Taking into account that image registration is an ill-posed problem, the presented approach is based on a variational formulation and particular emphasis is given to regularization functionals motivated by mathematical elasticity. Starting out from one of the most commonly used linear elastic models, its limitations and extensions to nonlinear regularization functionals based on the theory of hyperelastic materials are considered. A detailed existence proof for hyperelastic image registration problems illustrates key concepts of polyconvex variational calculus. Numerical challenges in solving hyperelastic registration problems are discussed and a stable discretization that guarantees meaningful solutions is derived. Finally, two case studies highlight the potential of hyperelastic image registration for medical imaging applications.