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We discuss recent advances in diffeomorphic image registration and related correspondence and shape matching problems. Diffeomorphic image registration is a classical, ill-posed, non-linear, non-convex, inverse problem with numerous applications in imaging sciences. It typically involves an infinite number of unknowns, which, upon discretization, results in high-dimensional, ill-conditioned systems. Image registration poses significant numerical challenges. We will showcase state-of-the-art techniques in scientific computing to tackle these challenges, highlight new theoretical developments, and discuss challenging application scenarios that require tailored formulations to obtain plausible solutions.