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A connecting thread between all these tasks, and a source of numerous challenges, is the inherent uncertainty in perceptual data. This dissertation considers how to manage rather than ignore perceptual uncertainty, enabling the creation of safe, robust, and reliable systems that will allow robots to leave the lab and enter the world. This dissertation introduces four novel approaches for perceptual uncertainty quantification. The first is a means of computing geometric uncertainty for radiance field models. Second, we propose a framework for estimating semantic uncertainty using vision and, third, we build off this framework to enable robots to use multiple sensing modalities such as vision and touch to update this semantic uncertainty. Lastly, we propose a means of propagating uncertainty over Riemannian manifolds with guarantees on numerical accuracy and with applications for robotic state estimation. 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