Abstract Image fusion or registration is central to many challenges in medical imaging today and has a vast range of applications. The purpose of this paper is to give an introduction to intensity based non-linear registration and fusion problems from a variational point of view. To do so, we review some of the most promising non-linear registration strategies currently used in medical imaging and show that all these techniques may be phrased in terms of a variational problem and allow for a unified treatment. A generic registration or fusion method depends on an appropriate chosen distance measure, a regularization, and some additional constraints. The idea of constraints is to incorporate higher level information about the expected deformation. We examine the most common constraints and show again that they may be conveniently phrased in a variational setting. As a consequence, all of discussed modules allow for fast implementations and may be combined in any favorable order. We discuss individual methods for various applications, including the registration of magnetic resonance images of a female breast subject to some volume preserving constraints.