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We present Splat-Nav, a real-time robot navigation pipeline for Gaussian splatting (GSplat) scenes, a powerful new 3-D scene representation. Splat-Nav consists of two components: first, Splat-Plan, a safe planning module, and second, Splat-Loc, a robust vision-based pose estimation module. Splat-Plan builds a safe-by-construction polytope corridor through the map based on mathematically rigorous collision constraints and then constructs a Bézier curve trajectory through this corridor. Splat-Loc provides real-time recursive state estimates given only an RGB feed from an on-board camera, leveraging the point-cloud representation inherent in GSplat scenes. Working together, these modules give robots the ability to recursively replan smooth and safe trajectories to goal locations. Goals can be specified with position coordinates, or with language commands by using a semantic GSplat. We demonstrate improved safety compared to point cloud-based methods in extensive simulation experiments. In a total of 126 hardware flights, we demonstrate equivalent safety and speed compared to motion capture and visual odometry, but without a manual frame alignment required by those methods. We show online replanning at more than 2 Hz and pose estimation at about 25 Hz, an order of magnitude faster than neural radiance field-based navigation methods, thereby enabling real-time navigation.