Abstract: The theory of Optimal Transport (OT) has developed rapidly during the last 30 years, partly because OT has constantly found applications in many areas, not only in mathematics, but also in economics and other social sciences. More recently a variant of OT, called Martingale Optimal Transport (MOT), has been introduced and extensively studied by mathematical finance community who observed the close connection of MOT with the model-independent robust option pricing and hedging. In this course we give a brief introduction of OT and MOT, from simple economic and financial motivations to some basic mathematical theory and open problems.