We present our work on scalable, GPU-accelerated algorithms for diffeomorphic image registration. The associated software package is termed CLAIRE. Image registration is a non-linear inverse problem. It is about computing a spatial mapping from one image of the same object or scene to another. In diffeomorphic image registration, the set of admissible spatial transformations is restricted to maps that are smooth, one-to-one, and have a smooth inverse. We formulate diffeomorphic image registration as a variational problem governed by transport equations. We use an inexact, globalized (Gauss--)Newton--Krylov method for numerical optimization. We consider semi-Lagrangian methods for numerical time integration. Our solver features mixed-precision, hardware-accelerated computational kernels for optimal computational throughput. We use the message-passing interface for distributed-memory parallelism and deploy our code on modern high-performance computing architectures. Our solver allows us to solve clinically relevant problems in under four seconds on a single GPU. It can also be applied to large-scale 3D imaging applications with data that is discretized on meshes with billions of voxels. We demonstrate that our numerical framework yields high-fidelity results in only a few seconds, even if we search for an optimal regularization parameter.