abstract: The concept of stretched Brownian motion has been introduced as the unique solution to a dynamic formulation of the martingale transport problem. The name originates from the characteristic of mimicking as closely as possible the movement of a Brownian particle, while fitting the prescribed marginals. In this talk I will review fundamental results for the stretched Brownian motion, with particular attention to the fixed-point iteration scheme for its computation, for which I will provide convergence results. Based on joint work with A. Marini and G. Pammer.