A Schroedinger bridge is a stochastic process which provides with a probabilistic version of the displacement interpolation between probability measures. In this talk I will start by surveying the parallelism between the Schroedinger problem and the Monge-Kantorovich problem. Next, I will present some recently obtained results for the dynamics of Schroedinger bridges. In particular, I will discuss an equation for the marginal flow, quantitative bounds for the evolution of the marginal entropy, and provide some applications. Finally, I will outline some connections between the so called ``reciprocal characteristics” of a Langevin dynamics and the convexity of the Fisher information.
