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The usage of neural networks (NNs) for flow reconstruction (FR) tasks from a limited number of sensors is attracting strong research interest, owing to NNs' ability to replicate high dimensional relationships. Trained on a single flow case for a given Reynolds number or over a reduced range of Reynolds numbers, these models are unfortunately not able to handle flows around different objects without re-training. We propose a new framework called Spatial Multi-Geometry FR (SMGFR) task, capable of reconstructing fluid flows around different two-dimensional objects without re-training, mapping the computational domain as an annulus. Different NNs for different sensor setups (where information about the flow is collected) are trained with high-fidelity simulation data for a Reynolds number equal to approximately $300$ for 64 objects randomly generated using Bezier curves. The performance of the models and sensor setups are then assessed for the flow around 16 unseen objects. It is shown that our mapping approach improves percentage errors by up to 15\% in SMGFR when compared to a more conventional approach where the models are trained on a Cartesian grid, and achieves errors under 3\%, 10\% and 30\% for pressure, velocity and vorticity fields predictions, respectively. Finally, SMGFR is extended to predictions of snapshots in the future, introducing the Spatio-temporal MGFR (STMGFR) task. A novel approach is developed for STMGFR involving splitting DNNs into a spatial and a temporal component. We demonstrate that this approach is able to reproduce, in time and in space, the main features of flows around arbitrary objects.