Optimal mass transport (OMT) provides a natural geometry for interpolating distributions (displacement interpolation) and modeling flows. As such it has been the cornerstone of many recent developments in physics, probability theory, image processing, time-series analysis, and systems and control. An alternative framework, rooted in statistical mechanics and large deviations, is that of Schrödinger bridges (entropic interpolation) which can in fact be seen as a stochastic regularization of OMT. In spite of the importance of these two problems and the extensive amount of related work in recent years, computational aspects remain challenging in high dimensions. The purpose of the talk is to discuss the relation between the two and to present elements of a computational approach based on the Hilbert metric. The talk is based on joint work with Yongxin Chen and Michele Pavon.