Summary: In the recent literature, martingale inequalities have been emphasized to be induced by pathwise inequalities independently of any reference probability measure on the paths space. This feature is closely related to the problem of robust hedging in financial mathematics, which was originally addressed in some specific cases by means of the Skorokhod embedding problem. The martingale optimal transport problem provides a systematic framework for the robust hedging problem and, therefore, allows to derive sharp martingale inequalities. We illustrate this methodology by deriving the sharpest possible control of the running maximum of a martingale by means of a finite number of marginals.