Generative machine learning models have become a known quantity, both in the fields of High Energy Physics, as well as in machine learning in general. Nearly all generative models teach a neural network to map a random variable with a known probability density, like a Gaussian, to realistic physics data. However, a wide range of problems, such as simulation refinement or unfolding, require not the mapping of random noise to physical distributions, but instead require the mapping of physical distribution to physical distribution. To this end, we introduce a novel type of generative model: The Schrödinger Bridge. These bridges are specifically designed to morph one arbitrary distribution to another. We demonstrate two implementations of Schrödinger Bridges and use them to refine fast shower simulation to the quality of full simulation, as well as to achieve state-of-the-art unfolding performance.