Abstract: Denoising diffusion models have recently emerged as a powerful class of generative models. They provide state-of-the-art results, not only for unconditional simulation, but also when used to sample from complex posterior distributions arising in a wide range of inverse problems such as image inpainting or deblurring. A limitation of these models is that they are computationally intensive as obtaining each sample requires simulating a non-homogeneous diffusion process over a long time horizon. We show here how a a Schrodinger bridge formulation of generative modeling leads to a theoretically grounded algorithm shortening generation time which is complementary to other proposed acceleration techniques. We further extend the Schrodinger bridge framework to perform posterior simulation. We demonstrate this novel methodology on various applications including image super-resolution and optimal filtering for state-space models.