Score-based generative models exhibits state of art performance on density estimation and generative modeling tasks.These models typically assume that the data geometry is flat, yet recent extensions have been developed to model data living on Riemannian manifolds. Existing methods to accelerate sampling of diffusion models are typically not applicable in the Riemannian setting and Riemannian score-based methods have not yet been adapted to the important task of interpolation of datasets. To overcome these issues, we introduce \emph{Riemannian Diffusion Schr\"odinger Bridge} (RDSB).Our proposed method generalizes Diffusion Schr\"odinger Bridge introduced in \cite{debortoli2021neurips} to the non-Euclidean setting and as such generalizes Riemannian score-based models beyond the first time reversal. We validate our proposed method on synthetic data and real Earth and climate data.