Image registration is an important tool in image analysis to spatially align pairs of images. In medical image analysis, for example, registrations to a common atlas space are frequently computed for population-based analyses. Image registration approaches typically consist of a suitable model of deformation and a measure of image similarity. Deformation models range from simple affine transformations to general diffeomorphic transformations, allowing for fine-grained local deformations of space. While diffeomorphic transformations are often desired they may be costly to compute. For example, while the large displacement diffeomorphic metric mapping (LDDMM) model results in diffeomorphic transformations it requires the optimization over high-dimensional spaces. Recently there has been a push in the image registration community (both in computer vision and medical image analysis) to learn aspects of these deformations. This talk will focus on regression approaches to predict deformations between images rapidly, while guaranteeing smoothness of the resulting transformation. If time permits, some approaches to learn similarity measures as well as to assess uncertainties in image registration will also be discussed.