In the first half of this talk I will give an overview of the recently developed theory of Martingale Optimal Transport (MOT), particularly highlighting the differences with the more classical theory of Optimal Transport (OT). For illustration: In a dynamical version of OT particles optimally follow deterministic (often 'straight') paths, whereas for MOT randomness and branching is the rule. Among the milestones of OT theory the contributions by Benamou, Brenier and McCann (among others) regarding the time-dependent version of the problem stand out. In the second half of this talk I will discuss some recently developed martingale counterparts of these results. In particular, I will provide an answer to the question: what is the least-diffusive martingale connecting two given measures?. This is based in joint work with M. Beiglböck, M. Huesmann and S. Källblad.
