Cardiovascular simulations are computationally expensive, especially in the context of patient-specific and physiological settings, where numerical methods need to handle robustly complex flow regimes. In particular, when addressing inverse problems, e.g., for the estimation of relevant biomarkers from medical images, the cost of forward simulations remains a major bottleneck toward clinical and pre-clinical applications. Projection-based reduced-order models and purely data-driven surrogate models have the potential to reduce the computational costs, and many examples of applications have been proposed in the past decade. However, in many cases, the potential of model-order reduction is strongly affected by geometrical variability. When handling anatomical shapes, which cannot be described by a low number of geometrical parameters (as could be the case in other engineering applications), the classical approaches of splitting the solution in a costly offline and an efficient online phase is not directly applicable. Focusing on the case of aortic geometries, this talk presents a recent approach for arbitrary shape registration, based on a ResNet extension of the classical LDDMM framework, which is adapted to handle efficiently large computational meshes, via a cost functional that accounts for mesh boundaries and a multigrid approach during the training phase. The registration algorithm delivers, for an arbitrary pair of surface meshes describing the ascending aorta, a diffeomorphic map between the point clouds defining the source and the target surface meshes. This map allows to register simulation results from different patients onto a common \textit{reference} shape, constructing a large manifold of solutions for further application in projection-based reduced-order modeling and for the training of graph neural networks. At the same time, the registration allows to push-forward reduced models on new domains, enabling the solution of patient-specific inverse problems on reduced spaces without the need of a new offline stage. We present detailed assessment of the registration approach for a cohort of synthetic aortic geometries created from a patient database, and discuss further ongoing applications towards generative shape modeling and domain uncertainty quantification.